UC-NRLF 


LIBRARY 

OF    THE 

UNIVERSITY  OF  CALIFORNIA 

GIFT    OK 


Received        <c&&&r:         ,  i8 
A  ccessions  No  . 


QUESTIONS 


GENERAL    PHYSICS 


IN    FOUR    PARTS 


Based  on  the  tenth  edition  of  Everett's  Translation  of 
DESCHANEL'S  NATURAL  PHILOSOPHY 


PART    I 


STATICS 


INCLUDING 


MECHANICS,   HYDROSTATICS  AND   PNEUMATICS 


BY  HAROLD  WHITING,  PH.D. 

Associate  Professor  of  Physics  in  the  University  of  California 


BERKELEY,  CALIFORNIA 

i»93 


COPYRIGHT,  1893 
BY  HAROLD  WHITING 


TABLE  OF  CONTENTS. 


PART  I.— STATICS. 


(A)  MECHANICS,  OR  THE  STATICS  OF  SOLID  BODIES. 

PAGE 

Introductory  -  i 

I.     Units  of  Measurement  3 

Forces  and  Displacement. 

II.     Crude  Ideas  of  Force  4 

III.  Force  Related  to  Displacement  5 

IV.  Gravitation  Units  of  Force  -  6 
V.     Absolute  Units  of  Force  7 

VI.     Specification  of  a  Force  8 

VII.     Composition  and  Resolution  of  Forces  Meeting  at  a  Point  9 

VIII.     Equilibrium  of  Forces  Meeting  at  a  Point  12 

IX.     Action  and  Reaction  13 

Forces  Without  Definite  Centres  or  Points  of  Application. 

X.     Transmissibility  of  Force  15 
XI.     Equilibrium    of    Forces  Acting  at  Two  Points   (Struts 

and  Ties)  -  16 

XII.     Translation  and  Rotation  16 

XIII.  Composition  and  Resolution  of  Couples  in  the  same  Plane  1 7 

XIV.  Composition  of  a  Force  and  a  Couple  in  the  same  Plane  19 
XV.     Composition  of  Parallel  Forces       -  20 

XVI.     Equilibrium  of  Three  Parallel  Forces  21 

XVII.     The  Lever  22 
XVIII.     Composition    and    Equilibrium   of  Couples   in    Parallel 

Planes  23 
XIX.     Composition  and  Resolution  of  Couples  in  Non-Parallel 

Planes  24 

XX.     Wrenches  25 

XXI.     Elasticity  of  Solid  Bodies     -                                                    -  27 


Forces  having  Definite  Centres  or  Points  of  Application. 

XXII.  Centre  of  Gravity 

XXIII.  Centre  of  Gravity  of  Geometrical  Figures 

'   XXIV.  Centre  of  Gravity  of  Suspended  Bodies 

XXV.  Stability  and  its  Relations  to  Centre  of  Gravity    - 

XXVI.  Sensitiveness  of  a  Balance,  as  related  to  Centre  of  Gravity 

Work. 

XXVII.  Work,  and  its  Relations  to  Centre  of  Gravity 

XXVIII.  Work  as  a  Criterion  of  Stability 

XXIX.  Principle  of  Work  applied  to  Mechanical  Powers 

XXX.  Conservation  of  Work 

XXXI.  Loss  of  Work  by  Friction    - 

XXXII.  Efficiency 

XXXIII.  Power 


PAGE 
29 

31 

33 
35 

37 


39 
4i 

42 

45 
46 
48 
49 


(£)  HYDROSTATICS,  OR  THE  STATICS  OF  FLUID  BODIES. 

Properties  of  Fluids. 

XXXIV.  Solids  and  Fluids  Distinguished 

XXXV.  Liquids  and  Gases  Distinguished    - 

XXXVI.  Fundamental  Assumptions  in  Hydrostatics 

XXXVII.  Conditions  of  Equilibrium  in  Fluids 

Hydrostatic  Pressure. 

XXXVIII.     Pressure 
XXXIX.     Balancing  Columns   - 
XL.     Atmospheric  Pressure 

XLI.     Flow  of  Liquids  Affected  by  Air  Pressure 
XLII.     Pumps 

Buoyancy. 

XLHI.  Centres  of  Pressure  and  Buoyancy 

XLIV.  Flotation 

XLV.  Principle  of  Archimedes 

XLVI.  Atmospheric  Buoyancy 

XLVII.  Apparent  Specific  Gravity    - 

Pneumatics. 

XLVIII.     Law  of  Avogadro 

XLIX.     Molecular  Theory  of  the  Pressure  of  Gases 
L.     Law  of  Boyle  and  Mariotte 
LI.     Unequal  Compressibility  of  Gases 
LII.     Vapors  and  Gases  Distinguished     - 


50 
5i 
54 
55 


57 
60 
62 
64 
65 


67 

68 
70 
72 
74 


76 

77 
78 

79 
80 


END  OF  PART  I. 


[UHIVBRSITtf 


QUESTIONS  ON  GENERAL  PHYSICS. 


INTRODUCTORY. 

The  following  questions  have  been  prepared  for  the  use  of 
college  students  in  connection  with  a  lecture  and  laboratory 
course  in  General  Physics.  The  asterisk  (*)  indicates  topics  with 
which  the  student  should  be  already  familiar  in  his  preparatory 
work.  The  dagger  (f)  indicates  topics  to  be  treated  or  supple- 
mented in  the  course  of  lectures.  The  double  dagger  (J)  indicates 
topics  illustrated  by  experiments  either  in  the  lecture-room  or  in 
the  laboratory.  Questions  containing  problems  to  be  solved  by 
the  student  are  marked  "  Prob."  References  are  as  follows  : 
D.,  to  sections  in  DESCHANEL'S  Natural  Philosophy,  translated 
by  EVERETT,  tenth  edition  ;  L,. ,  to  laboratory  exercises  ;  Q.,  to 
questions  in  the  list  immediately  preceding  the  reference,  unless 
otherwise  indicated  by  section  or  number.  References  which 
are  incomplete,  and  explanatory  remarks,  which  may  be  omitted 
in  recitation,  are  enclosed  in  parentheses. 

i.f  State  certain  rules  for  expression  to  be  observed  in  note- 
books and  in  recitation. 

2 .  State  the  distinction  between  ' 4  Natural  "  (or  ' '  Physical  ' ' ) 
phenomena  and  other  phenomena.  In  which  does  observation 
play  the  more  important  part?  D.  i. 

3.f  Show  that  "Science,"  even  in  its  most  general  sense, 
necessarily  involves  classification  or  arrangement  of  some  sort. 


2  INTRODUCTORY. 

4.  Explain  the  principles  upon  which  Natural  (or  Physical) 
Science    has    been    divided    broadly   into  Natural   History    and 
Natural  Philosophy.     In  which  of  these  divisions  does  classi- 
fication, and  in  which  does  the  study  of  cause  and  effect  receive 
the  relatively  greater  amount  of  attention  ?     D.  i. 

5.  Upon   what   farther   principles   has   Natural  Philosophy 
been  divided  into  Astronomy,  Biology,  Chemistry,  and  Physics, 
(or  Natural  Philosophy  in  its  more  restricted  sense)  ?     D.  2. 

6.f  To  which  of  the  above  divisions  is  the  Experimental 
Method  virtually  restricted,  and  in  what  does  the  Experimental 
Method  consist  ? 

7.  Should  the  descriptive  portions  of  Astronomy,  Biology, 
and  Chemistry  (e.g.,  Mineralogy)  be  included  under  Natural 
Philosophy  or  under  Natural  History  ?  Q.  I.  4. 

8.f  Distinguish  ordinary  descriptive  physics,  whether  studied 
in  the  class-room  or  in  the  laboratory,  from  general  physics 
(whether  mathematical  or  simply  deductive),  on  the  one  hand 
and  from  truly  experimental  (inductive)  physics,  including 
physical  investigation  and  measurement,  on  the  other  hand. 

9.f  Show  that,  from  the  nature  of  the  subjects  allotted  to 
Physics,  this  science  is  able  to  carry  both  inductive  and  deductive 
methods  farther  than  any  other  science. 

io.f  What  faculty  is  particularly  trained  by  quantitative 
experiments  or  measurements  ? 

n.f  State  some  of  the  advantages  of  a  course  in  general 
and  experimental  physics.  Show  that  such  a  course  trains  the 
faculties  of  expression,  observation,  classification,  experimenta- 
tion, inference,  explanation,  and  judgment  (the  last  three  being 
the  most  important  branches  of  reasoning,  viz.,  inductive, 
deductive,  and  quantitative). 

i2.*tName  the  principal  branches  of  Physics.  Explain  the 
following  terms  and  their  mutual  relations  :  Dynamics,  Kinetics, 
Statics,  Mechanics,  Hydrostatics,  Pneumatics,  Thermics,  Optics, 
Acoustics,  Electricity  and  Magnetism.  D.  3  and  9. 


3  UNITS  OF  MEASUREMENT.  [I. 

i.*  Define  the  foot  and  the  metre. 

2.*  Explain  the  use  of  a  graduated  scale  for  measuring 
length. 

3.*  Define  the  pound  and  the  gram  as  units  of  mass.    D.  160. 

4.*  Explain  the  construction  and  use  of  a  set  of  weights 
(with  the  method  of  substitution). 

5.*  Define  the  (mean  solar)  second. 

6.*  Explain  the  use  and  construction  of  a  clock. 

7.*  Define  velocity,  and  state  in  what  units  it  is  expressed. 

8.*  Define  the  area  of  a  surface,  and  name  different  units  of 
area. 

9.*  Find  the  area  of  a  plane  rectangular  surface  of  length, 
/,  and  breadth,  b. 

10.*  Define  volume,  and  state  in  what  units  it  is  expressed. 

ii.*  Find  the  volume  of  a  block  of  length,  /,  breadth, 
£,  and  thickness,  /. 

12.*  Define  density,  and  give  an  example.     D.  160. 

13.*  Distinguish  absolute  and  relative  density.     D.  160. 

14.*  Find  the  density,  d,  of  a  body  of  mass,  m,  and  volume, 
v.  (D.  1 60.) 

15.*  Find  the  mass,  m,  of  a  body  of  density,  d,  and  volume, 
v.  D.  1 60. 

1 6.*  Find  the  volume,  v,  of  a  body  of  mass,  m,  and 
density,  d.  (D.  160.) 

1 7. *|  Discuss  the  experimental  determination  of  the  density 
(i)  of  a  solid,  (2)  of  a  liquid,  and  (3)  of  a  gas,  by  direct 
measurements  of  mass  and  linear  dimensions,  and  show  that 
the  same  fundamental  principles  apply  to  each  case.  (D.  162.) 

1 8.*  Define  the  pound  weight  and  the  gram  weight  as  units 
of  force  (in  latitude  45°).  D.  161. 

19.*  Explain  the  signification  and  use  of  the  prefixes,  mega-, 
kilo-,  hekto-,  deka-,  deci-,  centi-,  milli-,  and  micro-. 

20.*  Explain  the  signification  of  the  hyphen  in  such  words 
as  foot-pound,  kilogram-metre,  etc. 

21.*  Explain  the  signification  of  "per"  in  such  expressions 
as  pounds  per  square  inch,  or  centimetres  per  second. 


4  II. 

CRUDE  IDEAS  OF  FORCE. 

i.f  Show  that  the  sense  of  touch  is  closely  connected  with 
our  ideas  of  force. 

2.  What  connection  exists,  in  general,  between  force  and 
the  muscular  effort  necessary  to  produce  or  to  resist  it?  D.  7. 

3.f  Name  certain  cases  in  which  the  maintenance  of  a  force, 
like  continued  muscular  effort,  requires  the  continuous  expendi- 
ture of  energy  of  some  sort ;  and  where,  if  the  source  of  energy 
be  cut  off,  exhaustion  ensues,  and  the  force  ceases. 

4.f  What  is  meant  by  the  "fatigue"  of  metals,  etc.,  under 
continued  strain  ?  Would  a  perfect  solid  exhibit  such  effects  ? 

5.f  Are  we  justified  in  supposing  that  the  maintenance  of 
inanimate  forces  in  general,  like  the  maintenance  of  muscular 
tension,  in  the  absence  of  a  fresh  supply  of  energy,  necessarily 
involves  fatigue  or  exhaustion  to  any  considerable  extent? 
Illustrate  by  the  force  of  gravity  upon  the  moon. 

6.f  What  misconception  is  likely  to  arise  from  the  use  of 
muscular  tension  as  the  type  of  a  force  ? 

y.f  Does  the  modern  idea  of  force  contain  any  reference  to 
the  supply  of  energy  by  which  it  may  happen  to  be  maintained  ? 

8.f  What  restriction,  not  observed  by  earlier  writers,  exists 
in  the  modern  use  of  the  word  force  ? 

9.f  Criticise  the  expressions,  force  of  a  waterfall,  force  of  the 
wind,  force  of  gravity. 

ro.f  Consider  a  "push,"  a  "pull,"  or  a  "shove"  as  instances 
of  a  force.  What  does  each  imply  as  to  the  direction  of  the 
force  with  respect  to  the  agent ;  as  to  the  point  of  application 
of  the  force,  and  as  to  the  state  of  tension  or  compression 
produced  ? 

n.f  Criticise  the  definition  of  a  force  as  a  "push  or  a  pull." 
(HALL'S  Elementary  Ideas.)  To  what  extent  is  it  supposed 
that  molecular  forces  can  be  so  classified  ?  Is  it  thought  that 
electromagnetic  forces  fall  into  either  category  ? 


III. 

FORCE  RELATED  TO  DISPLACEMENT. 

i.f  What  in  general  is  the  effect  of  a  force  upon  a  point  in 
an  elastic  body  ? 

2.f  What  relation  exists  by  definition  between  the  direction 
of  the  force  and  the  direction  of  the  displacement  which  it 
produces  in  a  homogeneous  elastic  medium  ? 

3.f  What  obvious  quantitative  relation  exists  by  definition 
"between  the  magnitudes  of  several  forces  acting  in  a  given 
direction  and  the  magnitude  of  the  resulting  force  ? 

4.J  Describe  one  or  more  experiments  showing  that  the 
displacement  produced  by  two  or  more  forces  acting  at  a  given 
point  in  the  same  direction  is  the  sum  of  the  displacements 
produced  by  the  separate  forces  when  acting  alone. 

5.f  State  how  forces  are  measured  (relatively)  in  Statics,  and 
upon  what  experimental  evidence  the  accuracy  of  this  measure- 
ment depends. 

6.f  Explain  HOOKE'S  law,  ut  tensio,  sic  vis. 

7.  What  is  meant  by  a  "dynamometer?"     D.  7. 

8.  Name  some  form  of  dynamometer  in  common  use.     D.  7. 
9'*  Describe  an  ordinary  "spring  balance."     D.   7. 

10. %  Describe  the  calibration  of  a  "  spring  balance." 

ii.*  Explain  the  use  of  a  "  spring  balance"  for  the  measure- 
ment of  forces. 


IV. 
GRAVITATION  UNITS  OF  FORCE. 

1.  Distinguish    between    the   uses    of  the  word   "weight" 
in   the  sense  of  "mass"  and  in  the  sense  of  "force."     Which 
use  is  adopted  by  most  modern  writers?     D.    161. 

2.  What  name  is  given  to  such  units  of  force  as  the  pound- 
weight,  or  the  weight  of  a  kilogram  ?     D.  8. 

3.  What  are  meant  by  gravitation  units  of  Torce  ?     D.  8. 

4.f  Describe  an  experiment  with  an  ordinary  balance  showing 
that  the  weight  of  a  bod}*  at  the  top  of  a  high  tower  is  less  than 
at  the  foot  of  the  tower. 

5.f  What  inference  concerning  the  constancy  of  gravity  is 
drawn  from  the  behavior  of  clocks  controlled  by  gravity 
pendulums  when  transported  from  one  latitude  to  another  ? 

6.f  What  inference  concerning  the  constancy  of  the  action 
of  springs  is  drawn  from  the  behavior  of  chronometers  with 
balance-wheels  controlled  by  hair-springs  when  transported  from 
one  latitude  to  another  ? 

7-t  State  the  result  of  testing  the  weight  of  a  given  body  in 
different  latitudes  by  means  of  a  spring  balance. 

8.  Explain  the  result  of  testing  the  weight  (?)  of  a  given 
body   in  different  latitudes  by  means  of  an   ordinary  balance. 
D.  161. 

9.  In  an  unknown  latitude,  what  kind  of  balance  would  you 
prefer  for  measuring  "weight"  in  each  of  its  two  senses?     Q. 

10.  In  what  sense  of  the  word  weight  may  it  be  said   to  be 
constant,  and  in  what  sense  to  vary  with  the  latitude  ?     Q. 

11.  Is  the  variation  in  gravitation  units  of  force  with  latitude 
or  altitude  sufficient  to  impair  their  commercial  value?     D.  8. 


V. 
ABSOLUTE  UNITS  OF  FORCE. 

i.f  What  is  meant  by  an  "  absolute  unit  of  force  ?"  Explain 
by  the  action  of  such  a  unit  upon  a  spring  balance  in  different 
latitudes. 

2.f  Discuss  the  possibility  of  graduating  a  spring  balance  so 
as  to  indicate  the  magnitudes  of  forces  in  absolute  units. 

3.f  Name  the  absolute  units  of  force  adopted  in  the  English 
and  in  the  Metric  Systems. 

4.f  Define  the  "  poundal  "  in  terms  of  the  weight  of  a  pound 
in  latitude  45  degrees,  where  the  velocity  acquired  by  a  falling 
body  in  one  second  is  32. 172  feet  per  second. 

5.f  Define  the  "  dyne  "  in  terms  of  the  weight  of  a  "  gram  " 
in  latitude  45  degrees,  where  the  velocity  acquired  by  a  falling 
body  in  one  second  is  980.61  centimetres  per  second :  the 
equivalent  of  32.172  feet  per  second. 

6.f  Give  instances  of  familiar  forces  measured  in  poundals 
and  in  dynes. 

7.  In  latitude  45  degrees,  how  would  you  reduce  grams  to 
dynes  ?  pounds  to  poundals  ?  and  the  reverse  ?  Q. 

8.f  What  branch  of  Physics  has  been  greatly  simplified 
by  the  use  of  absolute  units  of  force  ? 


VI. 
SPECIFICATION  OF  A  FORCE. 

i.     Can   a   force  be   applied  practically   to   a   point?     Give 
reasons  for  your  answer.     D.   12. 

2.*  What  is  meant  by  the  point  of  application  of  a  force? 
D.  n. 

3.  What  is  meant  by  the  line  of  action  of  a  force  ?     Is  the 
line  of  action  of  a  force  synonymous  with  the  direction  of  the 
force?     D.  13. 

4.  *  What  is  meant  by  the  magnitude  of  a  force  ? 

5.  What  elements  are  necessary  for  the  specification  of  a 
force  ?     Q. 

6.  Correct  DKSCHANEI/S  statement   that   "a  force  is  com- 
pletely specified  when  its   magnitude,    its  point   of  application 
and  its  line  of  action  are  all  given. ' '     State  what  interpretation 
must  be  given  to  the  words  "  line  of  action  "   in  this  statement 
in  order  that  it  may  be  true,  and  whether  this  interpretation   is 
or  is  not  in  keeping  with  his  use  of  the  expression  ?     D.  1 1  and  13. 


VII. 

COMPOSITION    AND    RESOLUTION    OF    FORCES 
MEETING    AT   A    POINT. 

1. 1  Describe  an  experiment  showing  that  the  direction  and 
magnitude  of  the  displacement  produced  by  a  given  force  acting 
on  a  point  in  a  homogeneous  elastic  medium  is  independent  of 
the  forces  already  existing  at  that  point,  provided  that  these 
forces  are  constant  in  magnitude  and  in  direction. 

2.  Show    that     all    geometrical     propositions    relating    to 
displacements  apply  also  to  the  forces  which  these  displacements 
represent.     Q. 

3.  Two  forces  acting  on  a  point  produce  independently  the 
displacements,  AB   and   BC.     What  is   the   resulting   displace- 
ment ?     Q. 

4.  What  name  is  given  to  a  single  force  which  will,  acting 
alone,    produce   the   same   displacement    as   two   or   more   than 
two  other  forces  when  acting  together?     D.    15. 

5.  Define  the  resultant  of  two  or  more  forces.     D.  15. 

6.f  Show  that,  if  forces  are  measured  by  the  displacements 
which  they  produce,  the  resultant  of  two  forces  is  represented  in 
direction  and  magnitude  by  the  diagonal  of  a  parallelogram,  the 
sides  of  which  represent  the  two  forces  in  question.  Q. 

7.*  Describe  the  ''parallelogram  offerees,"  and  its  use  in 
finding  resultants.  D.  16. 

8.*  Show  that  two  forces,  represented  by  the  lines  AB  and  BC, 
have  a  resultant  represented  by  a  line,  AC,  in  the  same  plane  as 
AB  and  BC.  D.  14,  15. 

9.*  Explain  the  "triangle  of  forces,"  its  relation  to  the 
"  parallelogram  of  forces,"  and  its  use  in  finding  resultants. 
D.  16. 

10.     Show  that  any  number  of  forces,  AB,  BC,  CD,  .   .   .   YZ, 
have  a  resultant,  AZ.     D.  18. 


10  COMPOSITION    OF    FORCES.  [vil. 

n.f  Consider  whether  the  statement  in  the  last  question 
applies  to  forces  in  one  plane  only,  or  to  forces  in  any  plane. 

12.  Show  that  the  resultant  of  three   forces  represented  by 
the  lines,  AB,  BC,  and  CD,  is  represented  by  the  diagonal  of  the 
parallelepiped  constructed  on  these  three  lines.     D.  18. 

13.  Show  that  in  the  parallelepiped   of  forces,    the  order  of 
combination  is  indifferent.     D.  18. 

14.  Describe  the  use  of  the  parallelepiped  of  forces  for  finding 
resultants  of  three  forces.     D.  18. 

15.*  What  name  is  given  to  two  or  more  forces  which  together 
produce  the  same  displacement  as  a  single  force  of  given 
magnitude  and  direction?  D.  15,  21. 

1 6.*  Define  the  word  component,     D.  15,  31. 

17.  Show    that    if   the  magnitude   and    direction   of   a  force 
and  two  of  its  three  components  are  given,  the  magnitude  and 
direction  of  the  remaining  component  is  determined.     Q. 

1 8.  Show  that,  if  the  magnitude  and  direction  of  a  force  and 
the  direction  of  its  three  components  are  given,   the  magnitude 
of  each  component  is  determined.     Q. 

19.*  What  name  is  given  to  the  process  of  finding  the 
components  of  a  force  in  two  or  three  different  directions  ?  D.  31. 

20.  Show   that  a   force   may    in    general    be    resolved    into 
components,  if  these  components  are  unlimited  either  in  number 
or  in  direction,  in  an  infinite  number  of  ways.     Q. 

21.  Name  certain  circumstances  under  which   the  resolution 
of  a  force  into  components  leads  to  definite  results.     D.  31,  Q. 

22.*  State  certain  necessary  limitations  between  the  magnitude 
of  the  resultant  of  two  forces  and  (the  sum  and  difference  of) 
these  forces  ;  also  between  the  sum  and  the  resultant  of  any 
number  of  forces.  Q. 

23.  What  name  is  given  to  components  of  a  force  when  these 
are  at  right  angles  to  each  other  ?  and  to  the  process  of  resolution 
in  such  cases  ?  D.  32. 


II  COMPOSITION   OF   FORCES.  [VII. 

24.  What  kind  of  composition   and  what  kind  of  resolution 
is  understood  unless  otherwise  stated  ?     D.  32. 

25.  Show   that   a   force     can     have    three     and     only    three 
rectangular  components,  and  that  it  is  completely  specified  when, 
and    only    when,    these     three   components    are    explicitly    or 
implicitly  given.     Q. 

26.  Find  by  the  Pythagorean  proposition  the  relation  between 
a  force  and  its  three  rectangular  components.      Q. 

27.  A    point,    A,    is    displaced    from    A'   to    A",    find    (by 
rectangular  projection)  how  much  it  has  approached  an  infinitely 
distant  point,  B.     Prob. 

28.  What   is    meant    by    the    (rectangular)    component    of  a 
displacement    or  a  force   in   or  along  a  given  direction,  line,   or 
plane?     D.  32. 

29.  State  the  trigonometric  relation  between  a  force  AB,  and 
its  (rectangular)  component  along  the  line,  AC.     D.  32. 

30.  What  in  general   is  the  trigonometric  relation  between  a 
force  and  its  component  in  a  line  (or  plane)  making  the  angle  A 
with  a  line  parallel  to  the  direction  of  the  force?     D.  32. 

31.  Show  that  a  force  can  have  no  (rectangular)  component 
in  a  direction  at  right-angles  with  itself.     Q. 

32.  Show  that  the  component  of  a  displacement   in   a  given 
direction  is  determined   solely  by  the  component  of  the   force  in 
that  direction,  and    is  not  affected   by  either  of  the  other  two 
components.     Q. 

33.  Show  that  the  component   displacement  in  any  direction 
due  to  one  of  the  components  of  a   force   acting  alone   in  this 
direction  is  the  same  as  that  due  to  the  whole  force.     Q. 


12  VIII. 

EQUILIBRIUM  OF  FORCES  MEETING  AT  A  POINT. 

1 .  When  is  a  point  in  an  elastic  body  said  to  be  in  equilibrium 
under  the  action  of  forces  estimated  by  the  displacements  which 
they  produce  ?     D.  9. 

2.  Show  that  a  point  in  an  elastic  body  is  in  equilibrium  if 
the  resultant  of  all  the  forces  acting  upon  it  is  zero.     Q. 

3.f  State  the  result  of  plotting  lines,  AB,  BC,  CD,  DE,  etc., 
so  as  to  represent  in  magnitude  and  direction  forces  producing 
equilibrium  at  a  point. 

4-f  What  is  meant  by  the  "polygon  of  forces?" 

5.*  When  are  three  forces  acting  at  a  point  in  equilibrium  ? 
D.  14. 

6.*  Show  that  three  forces  acting  at  a  point  cannot  be  in 
equilibrium  unless  they  are  contained  in  the  same  plane.  Q. 

7.*  Explain  the  "triangle  of  forces"  as  applied  to  forces  in 
equilibrium.  D.  14. 

8.J  Explain  the  determination  of  the  weight  of  a  body  by 
means  of  two  spring  balances  with  cords  meeting  in  a  knot  from 
which  the  body  is  suspended,  and  illustrate  by  examples.  L. 

9. 1  Explain  the  determination  of  the  weight  of  a  body 
suspended  by  a  cord  by  means  of  a  spring  balance  acting  upon  a 
knot  in  this  cord,  so  as  to  produce  a  known  angular  deflection.  L. 

10.     How  do  you  find  the  relative  magnitudes  of  three  forces 
of  known  direction,  producing  equilibrium  at  a  point? 

n.t  What  is    meant   by   the    "  equilibrant "    of  a   number  of 
forces  ? 

i2.f  Find  the  equilibrant  of  the  forces,  AB,  BC,   .    .   YZ.     Q. 

13.  Find  the  equilibrant  of  two  furces,  AB  and  BC,  and  show 
that  it  lies  in  the  plane  of  these  forces.     D.  14. 

14.  Show  that  in  the  triangle  of  forces  the  forces  and   their 
equilibrant  represent  rotatiori  in   the  same  direction  about  any 
point  in  the  interior  of  the  triangle.     D.  14. 

15.  Show  that  the  equilibrant  of  any  number  of  forces  is  in 
all  cases  equal  and  opposite  to  their  resultant.     Q. 


i3  IX. 

ACTION  AND  REACTION. 

1.  A  small   light  spring  of  the  normal    length,   A'  B' ,    is 
stretched  to  the  new  length,  A"  B" .     Find  the  displacement  of 
B  relatively   to  A,  and   the  displacement  of  A   relatively   to  B, 
(ist)  taking  AB  as  the  positive  direction,  and  (2d)  taking  BA  as 
the  positive  direction.     What  obvious  relations  exist  between  the 
magnitudes  and  between  the  signs  of  these  four  displacements? 
Prob. 

2.  Show  that   a  force,  if  represented  by  a  relative  displace- 
ment, necessarily  exists  in  four  different  aspects.      Q.   i. 

3.f  When  a  small  light  spring  is  stretched  from  the  normal 
length,  A  B' ,  to  a  new  length,  A"  B" ,  what  displacement  is  taken 
as  a  measure  of  the  force  exerted  upon  the  spring  by  B  ?  by  the 
spring  at  B  ?  upon  the  spring  at  A?  by  the  spring  at  A  ? 

4.f  Consider  the  modifications  (if  any)  which  would  be 
introduced  into  your  answers  to  the  questions  above  by  the 
substitution  of  compression  for  extension. 

5.  What  relation  exists  by  definition  between  the  forces 
exerted  at  a  given  point  upon  and  by  a  given  thing  (or  agent)?  Q. 

6.*  Considering  the  force  exerted  by  a  spring  upon  a  fixed 
support  as  the  "  action  "  of  the  spring,  what  name  is  given  to  the 
force  exerted  by  the  support  upon  the  spring  ? 

7.*  Considering  the  force  exerted  by  any  agent  (the  hand,  for 
instance),  upon  a  spring  as  the  "action"  of  that  agent,  what 
name  is  given  to  the  force  exerted  by  the  spring  upon  the 
agent  ? 

8.*  If  by  the  action  of  a  spring,  AB,  we  mean  the  force 
exerted  by  it  at  A,  what  name  do  we  give  to  the  force  exerted  by 
it  at  B  ? 

9.*  If  by  the  action  of  any  agent  upon  a  spring,  AB,  we 
mean  the  force  exerted  upon  the  spring  at  A,  what  name  do  we 
give  to  the  force  exerted  upon  the  spring  at  B  ? 

io.f  Distinguish  action  and  reaction  exerted  at  a  point  from 
action  and  reaction  exerted  through  some  medium  (for  instance, 
a  spring)  at  two  different  points. 

n.f  Show  that  action  and  reaction  are  (in  each  case)  names 
for  two  different  aspects  of  one  and  the  same  stress. 


14  ACTION    AND    REACTION.  [iX. 

i2.f  Show  that  action  and  reaction  in  an  elastic  body,  being 
measured  by  relative  displacement,  are  equal  and  opposite 
whether  that  body  be  at  rest,  in  uniform  motion,  or  under 
acceleration. 

i3-f  Show  that  action  and  reaction  are  necessarily  simulta- 
neous. 

i4.*fState  the  principle  of  action  and  reaction. 
15.*  Give  instances  of  action  and  reaction.     D.  10. 

16.*  Correct  the  following  statement:  "every  action  is 
followed  (!)  by  an  equal  and  opposite  reaction."  Q. 

17.*  Contrast  the  scientific  use  of  the  word  "reaction  "  with 
its  popular  use,  exemplified  in  the  phrase,  "reaction  of  public 
opinion."  Q. 

1 8.*  Criticise  the  illustration  of  action  and  reaction  by  the 
successive  swings  of  a  pendulum.  Q. 

iQ.f  In  applying  the  principle  of  action  and  reaction  to  astro- 
nomical phenomena,  what  assumption  is  made  with  respect  to 
the  velocity  of  propagation  of  gravity  ?  Q. 

20.  Why  cannot  the  principle  of  action  and  reaction  be 
applied  indiscriminately  (as  in  D.  10)  to  forces  which,  like 
mechanical  stresses  (e.g.  the  tension  of  a  cord)  or  like  magnetic 
attractions  and  repulsions,  require  (in  some  cases)  a  perceptible 
amount  of  time  for  their  propagation  ?  Q. 

2i.f  Show  that  certain  objections  to  the  principle  of  action 
and  reaction  disappear  if  the  forces  in  question  are  applied  at  (or 
indefinitely  near)  the  same  place. 

22.*  Apply  the  principle  of  action  and  reaction  to  the  pressure 
between  a  body  and  the  table  upon  which  it  rests  ;  to  the  force 
exerted  by  the  hand  in  pulling  a  rope  ;  to  the  tension  between 
the  two  halves  of  a  rope  during  the  l '  tug  of  war  "  ;  to  the  pull 
of  traces  upon  a  whifBetree  in  starting  a  wagon,  etc.,  etc.  State 
accurately  what  constitutes  the  pair  of  eqral  rnd  opposite  forces 
in  each  case. 

23.  f  Show  that  the  action  and  reaction  between  two  bodies, 
though  equal  and  opposite,  being  applied  to  different  bodies,  do 
not  tend  to  produce  equilibrium  in  either,  considered  by  itself. 


15 

X. 
TRANSMISSIBILITY  OF  FORCE. 

i.f  Describe  an  experiment  with  a  chain  of  light  spring 
balances  illustrating  the  transmission  of  force. 

2.f  Prove,  by  the  principle  of  action  and  reaction,  that  force 
is  transmitted  along  a  chain  of  light  spring  balances  without 
loss. 

3.f  Show,  by  considering  a  medium  transmitting  force  as  a 
sort  of  chain,  that  the  equality  of  action  and  reaction  can  be 
predicated  for  every  point  in  the  medium. 

4-f  Describe  one  or  more  cases  of  the  transmission  of  force 
between  two  points  in  a  light  rigid  body. 

5.J  Describe  one  or  more  experiments  illustrating  the  trans- 
mission of  force  by  a  cord,  or  other  flexible  substance. 

6.*  Criticise  the  reason  usually  assigned  (loss  of  force)  for 
hitching  a  horse  as  close  as  possible  to  his  load.  Q. 

7.  What  relation  exists  between  the  forces  at  various  points 
of  a  line,  AB,  any  one  of  which  would  be  neutralized  by  a  given 
force  at  A,  in  the  line,  AB  ?  (Axiom). 

8.f  Show  that  a  force  of  given  magnitude  in  the  direction, 
AB,  may  be  transferred  to  any  point,  B,  in  its  line  of  action 
without  modifying  the  result.  D.  13. 

9. \  Describe  one  or  more  experiments  with  a  rigid  body  held 
by  elastic  forces,  or  otherwise,  so  as  to  be  sensitive  to  any  change 
in  the  magnitude,  as  well  as  in  the  direction  or  line  of  action  of 
a  force,  and  state  the  effect  of  changing  the  point  of  application 
of  a  given  force  along  its  line  of  action. 

10.*  What  is  meant  by  the  transmissibility  of  a  force  along 
its  line  of  action  ?  D.  13,  Q. 

n.f  Show  that  the  line  of  action  of  a  force  may  be  substituted 
for  its  point  of  application  in  all  cases  involving  only  a  single 
position  of  a  body,  and  not  concerning  the  displacement  of  the 
point  itself.  Q. 


i6  XI.  [xn. 

EQUILIBRIUM  OF  FORCES  ACTING  AT  TWO  POINTS. 

i.     What  is  meant  by  a  "rigid  body"  in  mechanics  ?     D.  12. 

2.f  What  is  meant  by  a  "  strut  "  or  a  "  tie  "  ?  and  what  kind 
of  force  is  each  most  suitable  for  transmitting  ? 

3.f  What  relation  is  found  (experimentally)  to  exist  between 
the  forces  transmitted  by  a  strut  or  a  tie,  and  the  line  joining  the 
points  of  application  of  these  forces  ? 

4.*  Show,  by  general  considerations  of  symmetry,  that  two 
equal  and  opposite  forces  applied  at  two  points,  A  and  B,  of  a 
rigid  body,  parallel  to  AB,  should  produce  no  displacement 
(either  of  translation  or  of  rotation)  in  the  body  as  a  whole. 

5.f  When  is  a  rigid  body  said  to  be  in  equilibrium  ?     D.  9. 

6.  State  the  (three)  conditions  of  equilibrium   of  two  forces 
applied  at  two  points  in  a  rigid  body.     D.  13. 

7.  State  the  conditions  of   equilibrium  of  any    number  of 
forces  with  a  common  line  of  action.     D.  13. 

XII.— TRANSLATION  AND  ROTATION. 

i.*  Distinguish  displacements,  (whether  produced  by  forces  or 
not)  into  two  classes,  according  to  whether  translation  or  rotation 
is  produced.  D.  5. 

2.f  Describe  an  experiment  with  a  globe  illustrating  the  fact 
that  all  displacements  are  resolvable  into  translation  combined 
with  rotation  about  one  or  more  axes. 

3.  Show  that  the  displacement  of  a  body  is  completely 
specified  by  that  of  the  centre  and  two  points  on  the  surface  of  a 
sphere  in  which  it  is  included.  Prob. 

4-t  Show  (by  drawing  great  circles  through  any  two  points, 
A  and  B,  on  the  surface  of  a  sphere,  before  and  after  rotation) 
that  there  must  ba  two  antipodal  points,  (the  intersections  of  the 
great  circles)  which  are  not  affected  by  any  given  displacement 
of  a  spherical  surface  about  its  centre. 

5.  Show  that  every  displacement  of  a  body  about  a  point  in 
it  must  have  a  definite  axis.     Q.  4. 

6.  Show  that  every   displacement  must  be  resolvable  into 
one  of  translation,  combined  with  one  of  rotation  about  a  given 
axis.     Q.  4,  5. 


17  XIII. 

COMPOSITION  AND  RESOLUTION  OF  COUPLES  IN 
THE  SAME  PLANE. 

i.*  Distinguish  a  push  or  a  pull  from  a  twist,  torsion  from 
tension  or  compression,  according  to  the  displacement  produced. 
D.  5  and  6. 

2.*  State  the  effect  of  two  equal  and  opposite  forces  applied 
at  the  points,  A  and  B,  at  right  angles  with  the  line,  AB,  in 
an  elastic  medium. 

3.*  What  combination  of  forces  is  necessary  to  produce  a 
twist  ? 

4.  What  name  is  given  to  a  pair  of  equal  and  opposite  forces 
having  different  lines  of  action  ?    to   any  combination  of  forces 
producing  a  twist  ?     D.  26,  27. 

5.  Define  the  word  "  couple,"  and  tell  what  is  meant  by  the 
"arm,"  the  "moment,"  the    "plane,"  and   the    "axis"    of  the 
couple.     D.  26,  27. 

6.f  Distinguish  right  and  left-handed  couples  :  also  similar 
and  opposite  couples. 

7-t  Show  that  two  equal  and  opposite  couples  in  the  same 
plane,  having  equal  forces  and  equal  arms  not  parallel,  give  rise 
to  a  pair  of  equal  and  opposite  resultants,  having  for  their 
common  line  of  action  a  diagonal  of  the  parallelogram  formed  by 
the  prolongation  of  the  component  forces,  and,  therefore,  neutralize 
each  other,  as  far  as  rotatation  or  twist  is  concerned. 

8.  Show  that  two  similar  couples  in  the  same  plane,  with 
equal  forces  and  arms,  being  capable  of  neutralizing  the  effect  of 
a  given  couple,  are  equivalent  to  each  other.  (Axiom). 

9.J  Describe  one  or  more  experiments  showing  that  the 
translation  or  rotation  of  a  couple  in  its  own  plane  does  not 
modify  the  effect  which  it  produces. 

10.  Find  the  resultant  of  two  similar  couples  with  equal  forces 
and  with  arms,  AB  and  BC,  in  the  same  straight  line.  Prob. 

n.  What  assumption  is  made  in  the  last  question  ?.s  to  the 
disappearance  of  two  equal  and  opposite  forces  at  a  point  ?  and 
what  postulate  (as  to  the  introduction  of  equal  and  opposite 
forces  at  a  point)  will  enable  you  to  resolve  the  resultant  couple 
back  into  its  components  ?  Prob. 


1 8  COMPOSITION  AND    RESOLUTION    OF   COUPLES.  [xill. 

i2.f  Show  that  a  couple  with  given  forces  and  arm,  A,  may 
be  resolved  into  A  couples  with  the  same  forces,  but  with  unit 
arms. 

I3.f  Show  that  a  couple  with  given  arm  and  forces,  F,  can  be 
resolved  into  ^couples  with  the  same  arm,  but  with  unit  forces. 

i4.f  Show  that  a  couple  with  forces,  /%  and  arm,  A,  can  be 
resolved  into  F  X  A  couples  with  unit  forces  and  arms. 

Q-  12,  13. 

15.  What  name  is  given  to  the  product  of  the  force  and  arm 
of  a  couple  ?  D.  26. 

i6.f  Describe  one  or  more  experiments  with  a  torsion 
apparatus  showing  that  the  effect  of  a  couple  is  proportional  to 
the  force,  if  the  arm  is  constant  ;  to  the  arm,  if  the  force  is 
constant ;  and  to  the  moment,  if  the  force  and  arm  both  vary. 
State  the  (empirical)  law  established  by  these  experiments. 

17.  State  the  necessary  relation  existing  between  the  moment 
of  a  couple  and  the  twist  produced  in  an  elastic  medium,  in  which 
a  given  couple  always  produces  a  given  additional  twist.       Prob. 

1 8.  Show  (by  resolution  into  unit  couples)  that  the  resultant 
of  any  number  of  couples  in  the  same  plane  is  a  single  couple 
in  that  plane,  with  moment  equal  to  the  algebraic  sum  of  the 
moments  of  the  components.     Q. 

ig.f  Show  that  a  couple  can  be  balanced  only  by  an  equal  and 
opposite  couple.  D.  27. 

20. J  Describe  one  or  more  experiments  (with  E.  H.  HALL'S 
apparatus)  illustrating  the  conditions  of  equilibrium  of  two 
couples  in  the  same  plane. 

2i.f  State  the  conditions  of  equilibrium  of  two  or  more  couples 
in  the  same  plane.  D.  27. 

22.  What  is  meant  by  the  moment  of  a  force  about  a  point? 
D.  24. 

23.")"  Show  that  the  moment  of  a  couple  is  equal  to  the 
difference  (or  algebraic  sum)  of  the  moments  of  its  two  forces 
about  any  point  in  its  plane. 

24.  f  Apply  the  principle  of  moments  about  a  point  to  the  case 
of  equilibrium  as  to  rotation  about  this  point. 


XIV. 

COMPOSITION  OF  A  FORCE  AND  A  COUPLE  IN  THE 
SAME  PLANE. 

1.  Show  that  a  change  in  the  line  of  action  of  a  force  is 
equivalent  to  the  introduction  of  a  couple  whose  moment  is  the 
product  of  the  force  and  the  distance  between  its  old  and  new 
lines  of  action,  and  whose  plane  contains  these  lines.     D.  28. 

2.  Find  the  direction  of  rotation  introduced  by  the  displace- 
ment of  the  points  of  application  of  forces  in  special  cases,  (e.g., 
a  right-handed  displacement  of  a  downward  force).     Prob. 

3.f  The  point  of  application  of  a  force  of  magnitude  and 
direction,  AB,  is  moved  through  a  distance,  AC,  at  right-angles 
to  AB ;  find  the  nature  of  the  action  which  brings  this  about, 
and  the  complete  specifications  for  this  action. 

4.f  Show,  conversely,  that  the  resultant  of  any  force  and  any 
couple  in  the  same  plane  is  a  single  force  in  this  plane,  equal  in 
magnitude  to  the  component  force,  parallel  to  this  force,  and  at  a 
(perpendicular)  distance  from  it,  equal  to  the  quotient  of  the 
moment  of  the  couple  by  the  magnitude  of  the  force. 

5.  How  would  you  construct  a  couple  equivalent  to  a  given 
couple,  so  as  to  show  that  its  effect  upon  a  given  force  in  its  own 
plane  is    simply  to  displace  the  line  of   action  of  this  force  ? 
D.  28. 

6.  Find  the  direction  of  the  displacement  produced  in  the 
points  of  application  of  forces  by  couples  in  special  cases,  (e.g., 
a  left-handed  couple  and  an  upward  force).     Prob. 


20 


XV. 
COMPOSITION  OF  PARALLEL  FORCES. 

1.  Show  that  the  transfer  of  two  equal  forces,  having  the 
same  direction,  to  a  line  midway  between  them  is  equivalent  to 
the  introduction  of  two  equal   and  opposite  couples,   and  hence 
does  not  affect  the  result.     Q.  XIV. 

2.  Show  that  the  resultant  of  two  equal  forces  in  the  same 
direction  is  equal  to  the  sum  of  these  forces,  acting  always  in  a 
line  midway  between  the  lines  of  action  of  the  components.     Q. 

3.f  Show  that  the  transfer  of  any  two  forces,  AX  and  BY, 
having  the  same  direction,  to  a  common  point,  P,  dividing  the 
line,  AB,  into  parts  such  that  AP  X  AX  is  equal  to  BP  X  BY, 
being  equivalent  to  the  introduction  of  equal  and  opposite  couples, 
does  not  modify  the  result. 

4«f  Extend  the  demonstration  called  for  in  the  last  question 
to  the  case  of  any  two  unequal  forces  in  opposite  directions. 

5.f  Show  that  the  resultant  of  any  two  forces  having  the 
same  direction  is  a  single  force  in  the  same  direction,  equal  in 
magnitude  to  the  sum  of  the  component  forces,  with  its  line  of 
action  in  the  same  plane  with,  and  between  the  lines  of  action 
of  these  forces,  and  distant  from  the  line  of  action  of  each  in  the 
inverse  proportion  of  the  magnitudes  of  these  forces.  D.  20. 

6.  Specify  the  resultant  of  two  unequal  parallel  forces  in 
opposite  directions.     D.  20,  Q. 

7.  Compare  the  moments  of  two  parallel   forces  about  any 
point  in  their  plane  with  the  moment  of  their  resultant  about 
this  point.     D,  22,  24. 

8.  Show   (by  successive  composition)   that  any  number  of 
parallel  forces,  whose  algebraic  sum  is  not  zero,   must  have  a 
single  force  for  their  resultant.     Prob. 


21 


XVI. 
EQUILIBRIUM  OF  THREE  PARALLEL  FORCES. 

i.*  Show  that  three  parallel  forces  cannot  be  in  equilibrium 
unless  one  is  opposite  to  the  other  two.  Q.  XV. 

2.*  Show  that  the  equilibrant  of  two  parallel  forces  must 
have  the  same  line  of  action  as  their  resultant.  Q.  XI. 

3.*  Show  that  one  of  three  parallel  forces  in  equilibrium  must 
be  equal  in  magnitude  to  the  resultant  of  the  other  two,  and 
hence  equal  to  the  algebraic  sum  of  the  other  two.  (Axiom). 

4.*  Show  that  the  equilibrant  of  two  forces  in  the  same 
direction  must  lie  between  them,  and  in  the  same  plane  with 
them.  Q. 

5.f  Show  that  the  line  of  action  of  the  equilibrant  of  two 
forces  in  the  same  direction  must  divide  the  distance  between 
their  lines  of  action  into  parts  inversely  proportional  to  the 
magnitudes  of  the  two  forces,  in  order  that  equal  and  opposite 
couples  may  be  produced,  and  that  this  condition  is  consistent 
with  those  above. 

6.  Compare  the  moment  of  two  parallel   forces  about  any 
point  in  their  plane  with  the  moment  of  their  equilibrant  about 
this  point.     Q. 

7.  Where  is  the  equilibrant  of  two  parallel  forces  in  opposite 
directions  ?  and  what  becomes  of  it  as  the  forces  become  more  and 
more  nearly  equal  ?     D.  26. 

8.*  State  the  (three)  conditions  of  equilibrium  of  three 
parallel  forces.  D.  19. 


22 


XVII. 

THE  LEVER. 

i.*  Describe  an  ordinary  (straight)  lever,  stating  what  is 
meant  by  the  "arms"  of  the  lever,  the  "fulcrum,"  (F) 
"power,"  (/>)  and  "weight,"  (W).  D.  56,  57. 

2.*  Distinguish  three  classes  of  levers,  stating  which  of  the 
three  forces  is  in  the  middle  in  each  case.  D.  56,  57. 

3.*  In  which  class  of  levers  is  the  weight  always  greater 
than  the  power  ?  always  less  ?  sometimes  greater  and  sometimes 
less  ? 

4.*  Which  force  in  each  class  of  levers  is  equal  to  the  sum  of 
the  other  two  ? 

5.*  State  the  law  of  levers.     Q.  XVI. 

6.f  Show  that  each  of  the  three  forces  in  equilibrium  in  the 
lever  is  proportional  to  the  distance  between  the  lines  of  action 
of  the  other  two. 

7.^  Describe  one  or  more  experiments  with  the  arithmetical 
lever,  and  state  what  points  each  illustrates.  D.  25. 

8.*  What  is  meant  by  a  bent  lever?     D.  57. 

9.  Distinguish  the  "arms  "  of  a  bent  lever  from  the  "  arms  " 
of  the  couples  which  are  in  equilibrium.  D.  57. 

10*  When  are  the   forces  acting  on   a  bent  lever   in   equili- 
brium ?     Q.  XVI. 

n.J   Describe  one  or  more  experiments  (with  E.  H.  HALL'S 
apparatus)  illustrating  the  equality  of  moments  in  the  bent  lever. 


23 

XVIII. 

COMPOSITION    AND    EQUILIBRIUM    OF    COUPLES    IN 
PARALLEL  PLANES. 

i.f  Describe  the  condition  of  the  rod  of  a  torsion  apparatus 
during  torsion. 

2.f  Show  that  a  rod  subject  to  torsion  can  offset  the  action  of 
a  couple. 

3.  When  a  rod  is  twisted  by  a  couple  acting  at  one  end  of  it, 
what  is  the  nature  of  its  reaction  against  the  agent  through 
which  the  couple  is  produced  ?     (Axiom). 

4.  Show   (by  considerations  of   symmetry)   that  a   uniform 
free  rod,    subject   to  torsion,    must  react  equally  at   both   ends 
against  agents  producing  the  torsion.     (Axiom). 

5.t  What  is  meant  by  a  "  shaft  "  or  "  axle,"  and  when  is  it 
in  equilibrium  ? 

6.f  Show  that  couples,  like  forces,  exist  under  four  aspects, 
and  obey  the  same  laws  of  action  and  reaction. 

7.f  Prove  that  a  couple  is  transmitted  without  loss  at  right- 
angles  to  its  plane,  from  one  section  of  a  body  to  another,  and 
hence  from  one  end  of  the  body  to  the  other. 

8.  What  is  meant  by  the  transmissibility  of  couples  along 
their  axes  ?  Prob. 

9.f  Show  that  the  plane  or  axis  of  a  couple  has  no  definite 
position  in  space,  but  is  determined  only  by  parallelism  to  a  given 
plane  or  axis.  Prob. 

io.f  Show  that  the  resultant  of  any  number  of  couples  with 
parallel  planes  is  a  single  couple,  with  plane  parallel  to  those  of 
the  components,  and  with  a  moment  equal  to  the  algebraic  sum 
of  the  moments  of  the  components.  D.  27,  Q.  XIII.  and  XVIII. 

1 1. 1  Describe  one  or  more  experiments  (due  to  E.  H.  HALL) 
illustrating  the  conditions  of  equilibrium  between  couples  in 
parallel  planes. 

i2.f  Show  that  a  couple  is  completely  specified  when  its 
moment  and  general  direction  of  rotation  are  known.  Prob. 


24 

XIX. 

COMPOSITION    AND    RESOLUTION    OF    COUPLES    IN 
NON-PARALLEL  PLANES. 

i.f  Find  the  locus  of  all  antipodal  points  in  a  given  spherical 
surface  at  which  tangential  forces  can  be  applied  so  as  to  produce 
a  couple  equivalent  to  a  given  couple. 

2.f  Find  two  antipodal  points  in  a  given  spherical  surface 
where  tangential  forces  can  be  applied  so  as  to  produce  two 
couples  equivalent  to  any  two  given  couples  in  non-parallel 
planes. 

3.f  Show  that  the  parallelogram  of  forces  can  be  applied  to 
the  composition  of  couples  in  non-parallel  planes. 

4-f  Prove  that  the  resultant  of  two  couples  in  non-parallel 
planes  is  a  single  couple  in  some  intermediate  plane. 

5.  Apply  the  Pythagorean  proposition  to  the  relation  between 
the  moments  of  two  component  couples  in  planes  at  right-angles, 
and  the  moment  of  their  resultant.  Prob. 

6.f  Show  that  a  couple  can  be  resolved  into  components,  the 
first  parallel  to  any  plane,  and  the  second  making  a  right-angle 
(or  any  other  angle)  with  the  first. 

7.  Find  the  trigonometric  relation  between  a  couple  and  its 
(rectangular)  component  in  any  plane,  making  the  angle  A  with 
the  plane  of  the  couple.  Prob. 

8,f  Find  the  component  of  a  couple  in  any  plane  whose 
normal  makes  an  angle,  A,  with  the  axis  of  the  couple.  Prob. 

9-t  What  is  meant  by  the  moment  of  a  force  about  a  given 
axis?  and  what  distinction  is  made  between  positive  and  negative 
moments  ? 

io.f  Show  that  the  algebraic  sum  of  the  moments  of  two  equal 
and  opposite  forces  about  any  axis  is  equal  to  the  component  of 
the  couple  in  a  plane  at  right- angles  to  this  axis. 

n.f  Show  that,  in  a  state  of  equilibrium,  the  algebraic  sum  of 
the  components  of  all  the  forces  along  any  axis,  and  also  the 
algebraic  sum  of  their  moments  about  this  axis  must  be  zero. 


XX. 

WRENCHES. 

1.  Show  that  every  force  acting  upon  a  body  (whether  part 
of  a  couple  or  not)  can  be  resolved  into  a  force  with  line  of  action 
passing  through  any  given  point,  P,  and  a  couple.     D.  28,  29  ; 
Q.  XIV.  i. 

2.  What  is  the  nature  of  the  resultant  in  the  last  question, 
of  all  the  forces  passing  through  a  single  point,  /*?     Of  all  the 
couples  ?     D.  29  ;  Q.  VII.,  XVIII.,  XIX. 

3«t  Show  that  all  possible  forces  acting  on  a  body  can  be 
resolved  and  recompounded  into  a  single  force  with  line  of 
action  passing  through  a  given  point  and  a  single  couple. 
D.  29;  Q.  2. 

4.f  Show  that  the  couple  in  the  last  question  can  be  resolved 
into  two  components,  one  at  right-angles  to  the  force,  the  other 
parallel  to  it,  and  state  the  result  of  combining  one  of  these 
components  with  the  force.  D.  29,  Q.  XIX.  6,  XIV.  4. 

5.f  Show  that  all  possible  forces  acting  on  a  body  (whether 
parts  of  couples  or  not),  are  equivalent  to  a  single  force  with 
definite  line  of  application,  and  a  single  couple  at  right-angles 
with  this  force.  D.  29,  Q. 

6.  What  name  is  given  to  a  force  combined  with  a  couple  at 
right-angles  to  it  ?     D.  29,  30. 

7.  Define   the   word    "wrench"    as   used    in    (theoretical) 
mechanics.     D.  29,  30. 

8.  Show  that  all  possible  combinations  of  forces  are  reducible 
to  a  wrench.     D.  29,  30  ;  Q. 

9-f  Distinguish  right  and  left-handed  wrenches,  according  to 
the  direction  of  rotation,  as  looked  at  in  the  direction  of  the 
force. 


26  WRENCHES.  [XX. 

io.f  Show  that  the  reaction  against  a  right  or  left-handed 
wrench,  being  equal  and  opposite  in  respect  to  both  force  and 
couple,  is  a  wrench  of  the  same  kind,  as  concerns  the  relation 
between  linear  and  rotatory  displacement. 

n.f  Distinguish  wrenches  farther  into  two  classes,  according 
to  whether  extension  or  compression  is  produced. 

I2.f  Discuss  the  wrenches  exerted  in  certain  familiar  mechan- 
ical processes  (such  as  driving  a  screw). 

13-f  Show  that  extension  in  one  part  of  an  elastic  system 
implies  compression  in  some  other  part. 

14.  In  what  respect  are  the  wrenches  exerted  by  two  different 
parts  of  an  elastic  system  on  each  other  similar,  and  in  what 
respect  dissimilar  ?  Q. 

i5.f  Discuss  the  longitudinal  and  torsional  effects  of  mutual 
wrenches  in  the  ordinary  spring  curtain-roller. 

1 6.  State  the  law  of  action  and  reaction  as  applied  to 
wrenches.  D.  30. 


27 


XXI. 

ELASTICITY  OF  SOLID  BODIES. 

i.f  Name  three  effects  of  a  wrench  upon  an  elastic  body. 

2.*  Distinguish  stretching,  bending,  twisting,  and  com- 
pression. 

3.  What    is    meant   by    elasticity  ?    by    limits   of    (perfect) 
elasticity?  by  a  .set  ?  or  permanent  strain  ?     D.  126. 

4.  Define  a  stress,    a  strain,  and  a  coefficient  or  modulus  of 
elasticity.     D.  128. 

5. 1  Describe  one  or  more  experiments  illustrating  effects  of 
longitudinal  forces  upon  wires  or  rods. 

6.f  State  the  law  connecting  the  stretching  of  a  rod  with  the 
force  applied,  with  the  length  of  the  rod,  and  with  its  area  of 
cross-section. 

7.  What  is  meant  by  YOUNG'S  modulus  of  elasticity? 
D.  128. 

8.f  Give  some  idea  of  the  relative  magnitudes  of  the  forces 
necessary  to  produce  a  given  stretch  in  rods  of  different  materials, 
but  of  the  same  dimensions. 

g.f  State  the  usual  effect  of  stretching  on  the  diameter  of  a 
rod,  and  what  is  meant  by  "  POISSON'S  ratio." 

io.f  What  connection  (if  any)  exists  between  the  relative 
magnitudes  of  the  forces  necessary  to  produce  a  given  amount  of 
stretching,  and  the  relative  magnitudes  of  those  required  to 
produce  a  given  amount  of  bending  in  bodies  of  given 
dimensions  ? 

1 1. 1  Describe  one  or  more  experiments  relating  to  the  laws 
of  bending. 


28  ELASTICITY   OF   SOLID    BODIES.  [XXI. 

12. f  State  the  laws  connecting  the  bending  of  a.  beam  with 
the  load,  and  with  the  length,  breadth,  and  thickness  of  a  beam. 

134  Describe  one  or  more  experiments  relating  to  the  laws  of 
torsion. 

i4-t  State  the  laws  connecting  the  twisting  of  a  rod  with  its 
length,  with  its  breadth,  and  with  the  magnitude  of  the  couple 
applied. 

15. f  Give  some  idea  of  the  relative  magnitudes  of  the  couples 
necessary  to  produce  a  given  amount  of  twisting  in  bodies  of  the 
same  dimensions,  but  different  materials. 

i6.f  What  is  meant  by  a  modulus  or  coefficient  of  torsion  ? 

17.  What  is  meant  by  elasticity  of  volume?  by  resistance  to 
compression?  by  compressibility  ?  D.  129. 

i8.f  Explain  how  the  compressibility  of  solids  (as  well  as 
liquids)  can  be  demonstrated  by  an  (OERSTED'S)  piezometer. 

19. f  What  connection  (if  any)  exists  between  YOUNG'S 
modulus,  the  modulus  of  torsion,  and  the  resistance  to  compres- 
sion for  a  given  material  ? 

20.  Show  that  HOOKE'S  law  applies  to  cases  of  stretching, 
bending,  twisting,  and  compression. 


XXII. 
CENTRE  OF  GRAVITY. 

i.f  When  is  a  force  applied  to  a  rigid  body  said  to  have  a 
definite  point  of  application  ?  (D.  34). 

2.f  What  is  meant  by  a  centre  of  force?  by  the  centre  of  two 
or  more  forces  ? 

3-t  Distinguish  between  a  true  centre  of  force,  as  for  instance 
an  ideal  atom,  and  a  virtual  centre  of  force,  as  for  instance  the 
centre  of  Saturn's  rings. 

4.  Show  that  the  centre  of  two  parallel  forces  (not  equal  and 
opposite)  having  definite  points  of  application  lies  in   the  line 
joining  these  points  of  application.     D.  34. 

5.  Where  is  the  point  of  application  of  the  resultant  of  two 
parallel  forces  (not  equal  and  opposite)  ?     D.  34. 

6.  Show    that   if    any    number   of    parallel    forces    (whose 
algebraic  sum   is  not  zero)   have   definite  points  of  application, 
they  give  rise  to  a  resultant  with   a  definite  centre  or  point  of 
application.     D.  34. 

7.  Show  that  the   forces  which   the  earth's  gravity   exerts 
upon  the  molecules  of  a  small  body  at  the  earth's  surface,  having 
definite  points  of  application,    and  being  (practically)  parallel, 
must  have  a  definite  centre  or  point  of  application.     D.  33,  34. 

8.  What  name  is  given  to  the  centie  or  ]  cint  of  i  f  pl.'catic  n 
of  the  (parallel)   forces  which   gia\  ity  exerts  i  yon   the  different 
elementary  particles  of  a  body  ?     D.  34. 

9.  Define  the  term  "  centre  of  gravity,"  and  state  whether 
it   is  applicable    to   solids   only,   or   also  to    liquids  and  gases. 
D-  34.  35- 

io.f  Show  that  if  masses  are  measured  by  the  forces  which 
gravity  exerts  upon  them,  the  centre  of  gravity  coincides  (by 
definition)  with  the  "centre  of  mass." 


30  CENTRE    OF   GRAVITY.  [XXII. 

n.f  Would  the  statement  in  the  last  question  hold  if  the 
masses  were  not  measured  by  the  forces  which  gravity  exerts 
upon  them  ?  or  if  the  forces  in  question  were  not  parallel  ? 

12.  Under  what  circumstances  does  the  centre  of  (the  forces 
due  to)  attraction  coincide  with  the  centre  of  mass  ?     Q. 

13.  Under  what  conditions  does  the  resultant  of  two  or  more 
forces  have  a  definite  centre  or  point  of  application  ?     Q. 

14.  Where  is  the  centre  of  gravity   of  two  equal  masses?  of 
two  unequal  masses  ?     D.  34. 

15. f  Show  that  the  centre  of  gravity  of  two,  three,  or  four  equal 
masses  is  at  the  geometrical  centre  of  this  line,  triangle,  or 
pyramid  which  they  define.  (D.  37-39). 

i6.f  Show  that  three  or  more  unequal  masses  must  have  a 
fixed  centre  of  gravity,  independent  of  the  order  in  which  they 
are  considered.  (D.  37). 

17.  Show    (by    geometry)    that    in    a   system    of  mass,    M, 
consisting  of  two  masses,  m^  and  m2,  at  the  heights,  /i,  and  //2, 
the    height,    H,    of  the    centre    of    the    gravity    is  such   that 
MH=m^  h^  -j-  m2  h2.     D.  22,  45. 

1 8.  Extend  the  principle  of  the  last  question  to  the  case  of 
any  number  of  masses.     D.  23,  45. 

19.  Show  that  the  principle  of  the  last  two  questions  applies 
not  only  to  the  heights,  H,  h^  hz,  etc.,  of  the  centre  of  gravity 
and   separate   masses   composing   a   system,    but   also   to    their 
distances,  D,  d^  d2,  etc.,  from  any  plane.     D.  22,  23. 


XXIII. 
CENTRE  OF  GRAVITY    OF    GEOMETRICAL    FIGURES. 

1.  What  assumption   (as  to  the  distribution   of  weight)  is 
made  in  calculating  the  position  of  the  centre  of  gravity  of  bodies 
approximating  to  geometrical  figures  ?     D.  35. 

2.  What  is  meant  by  the  centre  of  gravity  of  a  geometrical 
solid?  surface?  or  line  ?     D.  35. 

3.*  Show  that  the  centre  of  gravity  of  a  unifoim  thin  rod  (or 
line)  is  at  its  centre,  and  that  it  will  balance  about  any  axis 
passing  through  this  centre.  D.  36. 

4.f  Show  that  the  centre  of  gravity  of  two  equal  triangles  or 
pyramids,  symmetrically  situated  with  respect  to  their  common 
apex,  is  at  this  apex. 

5.  Prove  that   if  a  figure  can  be  cut  up  into  strips,  slices, 
pairs  of  triangles,  pairs  of  pyramids,  etc.,  each  having  its  centre 
of  gravity  at  or  in  a  given  point  or  axis,  the  centre  of  gravity  of 
the  whole  figure  must  be  at  or  in  this  point  or  axis.     Prob. 

6.  Show  that  if  any  figure  is  symmetrical  with  respect  to  a 
centre  (so  that  it  can  be  cut  up  into  equal  and  opposite  pairs  of 
pyramids,  with  common  apex  at  this  centre)  its  centre  of  gravity 
must  lie  at  this  centre.      Prob.     D.  36.   Q.  XVIII. 

7.  Show  that  if  any  figure  is  symmetrical  with  respect  to -an 
axis,  (so  that   it  can  be  cut  up  into  equal   and   opposite  pairs  of 
triangles  with  common   apex  on  this  axis)  its  centre  of  gravity 
must  lie  on  this  axis.     Prob.      (Q.  XXIII.) 

8.  Show  that  if  any  figure  is  symmetrical  with  respect  to  a 
plane  (so  that  it  can  be  cut  up  into  uniform  thin  strips,  each  with 
its  centre  of  gravity  in  this  plane),  the  centre  of  gravity  of  the 
figure  must  lie  in  this  plane.     Prob.      (Q.  XXIII.) 

9.  Where  is  the  centre  of  gravity  of  a  figure  symmetrical 
with  respect  to  two  axes  ?  an   axis  and   a  plane  ?  two  planes  ? 
three  planes  ?     Prob. 


32  CENTRE  OF  GRAVITY  OF  GEOMETRICAL  FIGURES.     [XXIII. 

10.  Find  the  centre   of  gravity   of  a  circle,   square,   ellipse, 
parallelogram,     cube,     block,    parallelepiped,    prism,    cylinder, 
sphere,  ellipsoid,  etc.     Prob.     D.  36. 

11.  Show  (by  cutting  up  a  triangle  into  strips  parallel  to  the 
base,  and  considering  the  centre  of  gravity  of  each  strip)  that 
the   centre   of   gravity   of    a   triangle   must    lie   in   its    medial 
line.     D.  37. 

12.  Show  that  the  centre  of  gravity  of  a  triangle  must  lie  at 
the   point   of    intersection    of    its   medial   lines,    and    find    (by 
geometry)  the  relative  altitudes  of  the  apex  and  centre  of  gravity. 
D.  37,  Q.  ii. 

13.  Show  (by  cutting  up  a  pyramid   into  sections  parallel  to 
the  base)  that  if  a  line  be  drawn  from  the  apex  to  the  centre  of 
gravity  of  the  base  (so  as  to  pass  through   the  centre  of  gravity 
of  each  section),    this  line  will  contain  the  centre   of  gravity  of 
the  pyramid.     D.  38. 

14.  Show  that  the  centre  of  gravity  of  a  pyramid  must  lie  at 
the  point  of  intersection  of  two  or  more  lines  drawn  as  in  the 
last  question,  and  find   (by  geometry)   the  relative  altitudes  of 
the  centre  of  gravity  and  apex  of  a  triangular  (or  any  other) 
pyramid.     D.  38. 

15.  Find  the  centre  of  gravity  of  a  cone  by  the  same  method 
as  in  the  last  question.     D.  38. 


33 


XXIV. 

CENTRE  OF  GRAVITY  OF  SUSPENDED  BODIES. 

i.f  Find  the  moment  of  the  force  exerted  by  gravity  on  a 
body  of  weight,  W,  about  a  horizontal  axis  in  a  vertical  plane  at 
the  distance,  D,  from  the  centre  of  gravity.  (D.  58). 

2.f  A  body  of  weight,  W,  and  centre  of  gravity  at  the 
horizontal  distance,  D,  from  a  fulcrum  (or  axis)  is  balanced  by  a 
force,  P,  acting  on  a  horizontal  arm,  A.  Find  the  equal  and 
opposite  moments.  (D.  58). 

3.  In  the  last  question,  express  each  of  the  four  quantities, 
W,  Z>,  P,  and  A,  in  terms  of  the  other  three.  Prob.  Q.  3. 

4.J  How  can  you  find  the  weight  of  a  body  by  balancing  it 
on  an  axis  by  means  of  a  known  weight  ?  Q. 

5-J  How  can  you  locate  the  centre  of  gravity  of  a  body  of 
known  weight  when  balanced  on  an  axis  with  another  body  of 
known  weight  ?  Q. 

6.J  Show  how  the  weight  of  any  object,  great  or  small,  can 
be  found  by  a  graduated  lever  of  known  weight.  Q. 

7-1  Show  how  the  position  of  a  bullet  of  known  weight  in  a 
gun-barrel  of  known  weight  can  be  found  from  the  displacement 
of  the  centre  of  gravity.  Q. 

8. *£  Describe  one  or  more  experiments  with  levers  showing 
that  their  weight  acts  always  as  if  concentrated  at  their  centre  of 
gravity. 

9.*  Explain  and  describe  one  cr  mure  experiments  illustrating 
the  behavior  of  bodies  suspended  at  their  centre  of  gravity. 

10.  Show  that  a  body  is  in  equilibrium  if  suspended  at  any 
point  in  a  vertical  line  passing  through  the  centre  of  gravity. 
D.  41. 

IT.*  What  is  the  result  of  suspending  a  body  at  a  point  not 
contained  in  the  vertical  line  passing  through  the  centre  of 
gravity  ? 


34  CENTRE  OF  GRAVITY  OF  SUSPENDED  BODIES.  [XXIV. 

12.  State  the  condition  of  equilibrium  in  a  body  suspended 
at  a  point.  D.  41. 

13.*  Describe  the  experimental  location  of  the  centre  of 
gravity  of  a  body  by  the  method  of  suspension.  D.  43,  44. 

14.  J  Describe  and  explain  one  or  more  experiments  in  which 
a  body  is  suspended  upon  a  horizontal  axis  in  a  vertical  plane 
containining  the  centre  of  gravity,  and  state  how  the  centre  of 
gravity  may  be  located  by  such  experiments. 

15.*  What  is  the  result  of  suspending  a  body  upon  a  horizontal 
axis  not  containing  the  centre  of  gravity  ? 

1 6.  What  is  the  result  of  suspending  a  body  upon  an  oblique 
axis  containing  the  centre  of  gravity  ?  not  containing  it?  Prob. 

17.*  Describe  the  condition  of  a  body  suspended  about  a 
vertical  axis. 

1 8.  State  the  conditions  of  equilibrium  of  a  body  suspended 
upon  an  axis,  in  terms  of  the  position  of  the  centre  of  gravity 
with  respect  to  this  axis.     D.  41. 

19.  Show  that  in  all  the  cases  above,  where  a  body  is  not  in 
equilibrium,  the  action  of  gravity  is  such  as  to  cause  the  centre 
of  gravity  to  descend.     D.  16. 

20.  Show  that  a  heavy  body  is  in  equilibrium  when,  and  only 
when,  the  nature  of  its  restraints  is  such  as  to  prevent  the  centre 
of  gravity  from  descending.     Prob. 

2 1. 1  Explain  the  action  of  a  double-pointed  cone  in  rolling  up 
two  inclined,  but  diverging  rails. 

22.  Explain  the  action  of  a  ball   in  rolling  into  the  deepest 
portion  of  a  hollow.     D.  46. 

23.  Show   that,   in   general,   the  centre  of    gravity   tends    to 
descend.     D.  46. 


35 


XXV. 

STABILITY,    AND    ITS    RELATIONS    TO    CENTRE    OF 

GRAVITY. 

1.  Show  that  a   body,    in  which  two  points    are   fixed,    is 
equivalent  to  one  suspended  upon  an  axis  passing  through  these 
points.     Prob. 

2.  How  many  points  in   a  body  must  be  fixed   to  prevent 
movement  of  any  kind  ?     Prob. 

3.f  Three  points  in  the  same  horizontal  plane  are  fixed.  Find 
the  direction  of  the  force  on  each,  according  to  the  position  of 
the  centre  of  gravity. 

4.  Describe  the  forces  by  which  a  tripod  is  usually  held  in 
place,  stating  the  direction   of  these  forces,   and   whether  they 
vary  uniformly  or  "  per  saltum  "  under  displacement.    D.  40,  52. 

5.  State  the  conditions  under  which  a  body  may  be  fixed  by 
the  upward  reaction  of  three  forces.     Q. 

6.f  What  name  is  given  to  the  area  of  a  horizontal  section 
included  between  the  forces  in  the  last  question  ? 

7.*  Define,  in  general,  the  "area  of  support"  of  a  bod)T.    Q. 

8.f  Show  that  a  body  is  stable  if  its  centre  of  gravity  lies 
above  its  area  of  support.  (D.  40). 

9.  What  is  meant  by  the  limit  of  stability  in  a  body 
supported  at  three  or  more  points  ?  D.  54. 

10.  Show  that  a  body,  displaced  within  its  limits  of  stability, 
tends  to  return  to  its  original  position.     Prob. 

11.  When  is  a  body  said  to  be  "  practically  stable  ?"     D.   54. 

12.  What,  in  general,  is  meant  by  stability?     D.  42. 

13.  Distinguish    stable,    unstable,    and   neutral   equilibrium, 
according  to  the  tendency  of  a  body  when  displaced  to  return  to 
its  original  position  of  equilibrium.     D.  42. 


36  STABILITY  AND  CENTRE  OF  GRAVITY.  [XXV. 

14.  When  is  a  heavy  body,  suspended  by  a  point  or  by  an 
axis,  in  stable  equilibrium  ?  in  unstable  equilibrium  ?  in  neutral 
equilibrium?      State    the    position    of    the    centre    of    gravity 
relatively   to   the   point   or   axis   of    suspension   in    each   case. 
D.  41,  42. 

15.  Explain  the  toy  called  "  the  balancer."     D.  53. 

1 6.  Show  that,  in  a  condition  of  stability,  the  height  of  the 
centre  of  gravity  is  at  a  minimum,  in  a  condition  of  instability 
at  a  maximum.     D.  46. 

17.  State  the  conditions  which  determine  whether  equilibrium 
exists,  and  whether  it  is  stable  or  unstable,  in  terms  of  the  path 
of  the  centre  of  gravity  consequent  upon   a  given  displacement 
of  the  body.     D.  46. 

1 8.  Find   the  state  of   equilibrium  in   a  sphere  or  cylinder 
rolling  on  a  flat,  on  a  concave,  and  on  a  convex  surface.     Prob. 
(D.  46). 

19.  Describe  and  explain  the  action  of  a  toy  known  as   "  the 
tumbler,"  or  any  other  similar  device.     D.  53. 


37  XXVI. 

SENSITIVENESS   OF   A    BALANCE    AS    RELATED    TO 
CENTRE  OF  GRAVITY. 

i.     Name  the  essential  parts  of  a  balance.     D.  70. 

2.J  Describe  the  beam  of  an  ordinary  balance,  the  knife- 
edges  and  their  bearings.  D.  75. 

3.J  Describe  the  pointer  or  index  of  a  balance,  the  scale-pans, 
and  the  means  for  arresting  the  loads,  or  lifting  them  from  the 
knife-edges. 

4.  Explain  the  importance  of  preserving  the  sharpness  of 
the  knife-edges,  and   the  ordinary  precautions  taken  to  this  end. 
(D.  75). 

5.  Where  is  the  centre  of  gravity  of  a  balance-beam,  and  how 
is  the  position  of  the  centre  of  gravity  adjusted  ?     D.  73. 

6.  Show  that  the  weights  act  as  if  concentrated  at  the  knife- 
edges.     Prob.     Q.  XXII.  i. 

7.  Show  that  if  the  three  knife-edges  be  in  the  same  straight 
line,  the  centre  of  gravity  is  not  disturbed  by  loading  the  scale- 
pans  so  as  to  produce  equilibrium.     Prob.     Q.  XXII.  5. 

8.  Show  that   in   a  straight-arm   balance,    we   have  for  the 
angle,    a,    of  deflection  due  to   excess  of  weight,  p,   on  aim   of 
length,  /,  of  the  beam  of  weight,  w,  with  its  centre  of  gravity  at 
a  distance,  d,  from  the  axis, 

tan  a  —  pi  -j-  wd.  D.  73. 

9-f  What  additional  terms  must  be  considered  if  the  three 
knife-edges  are  not  in  the  same  straight  line  ?  D.  74. 

10.  Show  that  if  the  outer  knife-edge   be  above  the  middle 
knife-edge,  the  equilibrium  of  the  balance  may  become  unstable 
from  loading.     D.  74. 

11.  Show  that  if  the  outer  knife-edges  be  below  the  central 
knife-edge,    the  equilibrium   of  the  balance  will    become  more 
stable  on  loading.     Prob.     D.  74. 

12.  Distinguish   three  types  of  balance,   according  to  align- 
ment of  the  knife-edges.     Q. 

13.  Show  that  a  balance  beam  may  belong,  successively,  to 
the  three  several   types  named  in  the   last    question,    through 
bending  under  an  increasing  load.     Prob. 

i4.f  What  is  meant  by  the  sensitiveness  of  a  balance  ? 


38  SENSITIVENESS   OF  A  BALANCE.  [XXVI. 

15.  State  certain  conveniences  in  the  use  of  a  balance  of 
known  sensitiveness.  D.  74. 

i6.f  Describe  the  method  of  weighing  under  a  constant  load, 
and  state  some  of  the  advantages  and  disadvantages  of  this 
method.  D.  74. 

ly.J  How  is  the  sensitiveness  of  a  balance  under  a  given  load 
ordinarily  determined? 

i8.f  Explain  the  use  and  construction  of  a  table  showing  the 
sensitiveness  of  a  balance  under  different  loads. 

ig.J  Describe  the  use  of  a  "  rider,"  and  show  that  in  weighing 
with  a  rider,  the  sensitiveness  of  a  balance  need  not  be  kngwn. 

20. J  Explain  the  method  of  weighing  by  "oscillations"  (or 
by  "vibrations  "),  and  state  some  of  its  advantages. 

2i.f  Show  that  friction,  within  certain  limits,  does  not 
necessarily  affect  the  sensitiveness  of  a  balance,  if  the  method  of 
oscillations  is  employed. 

22. f  State  what  advantages  are  to  be  gained  by  using  a  balance 
with  rapid  oscillations,  to  what  extent  sensitiveness  should  be 
sacrificed  to  this  end,  and  to  what  extent  sensitiveness  is 
consistent  with  rapidity.  D.  71,  74. 

23. f  Show  that  it  is  convenient,  but  not  necessary,  that  the 
pointer  should  indicate  how  much  the  loads  differ. 

24.  Correct  DESCHANEL'S  statement  (§73)  that  the  centre  of 
gravity  and  axis  of  a  balance  "  must  not  coincide."     Q. 

25.  What  qualities  are  sought  for  in  a  balance?     D.   71,    73. 
Q.  XXVI.  1-26. 

26.*  Describe  the  ordinary  "steelyard,"  andshowthat  equality 
of  the  arms  of  balance  is  not  necessary  in  order  that  the  true 
weight  of  a  body  may  be  found.  D.  76. 

27.  Describe  (BORDA'S  method  of)  weighing  by  substitution, 
(or  double  weighing,  according  to  DESCHANEL  and  other  French 
writers).  D.  72. 

28. J  Describe  (what  is  meant  by  most  English  writers  by)  the 
method  of  "double  weighing,"  (or  GAUSS'S  method,  involving 
an  interchange  of  the  loads). 

29. f  Show  that  the  method  of  double  weighing  (by  inter- 
change) is  twice  as  accurate  as  weighing  by  substitution. 

30. J  Describe,  in  detail,  the  processes  actually  employed  in 
very  accurate  weighings. 

"-.'• 

Qg  THE 


39 


XXVII. 

WORK,  AND  ITS  RELATIONS  TO  CENTRE  OF 
GRAVITY. 

i.*  What  name  is  given  to  the  product  of  a  force  (when 
constant  in  magnitude  and  in  direction)  and  the  displacement  of 
its  point  of  application  (in  the  same  direction)  ? 

2.*  Define  work,  and  name  some  of  the  units  in  which  it  is 
measured. 

3.*  Define  the  foot-pound  and  kilogram-metre. 

4.f  Define  the  foot-poundal,  and  erg  (or  dyne-centimetre). 

5.*  Find  the  work,  W,  necessary  to  raise  a  weight,  w, 
through  the  vertical  distance,  d.  D.  45,  47. 

6.*  Distinguish  between  the  work  done  by  a  force  upon  a 
weight  or  against  gravity,  and  that  done  by  gravity  or  by  the 
weight  against  the  force.  D.  47. 

7.  How  is  work  measured  when  the  directions  of  the  force 
and  displacement  are  nut  the  <-ame  ?     D.  45,  48. 

8.  The  point  of  application,  A,  of  a  force,  AB,  moves  to  C. 
What  kind  of  work  is  done  by   the  force   if  BAC  is  acute?  if 
BAG  is  obtuse  ?  if  BAC  is  a  right-angle  ?     D.  48. 

9.  Find  the  work   done  by  a  force,  AB,    in   producing  the 
displacement,  AC,   in   terms  of  AB,  AC,  and  a  function  of  the 
angle,  BAC.     D.  48. 

10.  Show  that  the  work  done  by  a  force  in  producing  a  given 
displacement  may  be  measured  (i)  by  the  product  of  the  displace- 
ment and  the  component  of  the  force  in  the  direction  of  the 
displacement,  or  (2)  by  the  product  of  the  force  and  the 
component  of  the  displacement  in  the  direction  of  the  force. 
Prob.  Q. 


40  WORK  AND   CENTRE    OF  GRAVITY.  [XXVII. 

11.  Show  that  the  work  necessary  to  move  a  weight  in  any 
direction  is  the  product  of  the  weight  and  the  vertical  component 
of  the  displacement.     D.  48. 

12.  A  body  of  weight,  W,  remains  fixed,  while  another  body, 
of    weight,    w,    moves    through    the    distance,     D.       Find    (by 
geometry)  the  displacement,   dy  of  the  centre  of  gravity  of  the 
two  weights.     Prob. 

13.  Compare,    in  the   last  question,   the  components   of  the 
two  displacements  in  the  vertical  (or  any  other)  direction.    Prob. 

i4.f  Show  that  the  product  of  a  weight,  forming  part  of  a 
system,  by  its  vertical  displacement  is  equal  to  the  product  of  the 
weight  of  the  whole  system  by  the  vertical  displacement  of  its 
centre  of  gravity.  Q. 

i5-t  What  is  meant  by  the  work  done  upon  the  centre  of 
gravity  of  a  system  ? 

i6.f  Show  that  the  algebraic  sum  of  the  quantities  of  work 
done  upon  the  separate  weights  of  a  system  (by  considering  these 
one  by  one)  is  equal  to  the  work  done  upon  the  centre  of 
gravity  of  the  system.  Prob. 

17.  State  the  "principle  of  work  "  as  applied  to  the  centre  of 
gravity  of  a  system  of  weights.     D.  45. 

1 8.  Bxplain  how  the  calculation  of  work  done  upon  a  body 
can  be  simplified  by  the  principle  of  the  last  question.     D.  45. 

19.  Find  the  work,   W,  necessary  to  pile  a  given  weight,  w, 
of   fragments  into  a  rectangular,    into    a    triangular,    or  into  a 
pyramidal   heap  of  height,    h,    above  the   (mean)   level   of   the 
fragments.     D.  45. 

20.  Modify  the  last   question  so   as  to  apply  to  the  case  of 
filling   cisterns  ot  different  shapes  with  liquids,   or  exhausting 
wells  of  different  depths  and  shapes.     Prob. 


XXVIII. 
WORK,  AS  A  CRITERION  OF  STABILITY. 

1.  Find  the  work,   W,  necessary  to  turn  a  block  of  length,  /, 
breadth,    b,    thickness,   /,    and  weight,    w,    from   its  position  of 
maximum    stability  into  its    position  of    medium    or  minimum 
stability.     Prob.      D.  45. 

2.  Modify  the  last  question   so  as  to  apply  to  an  ellipsoid. 
Prob. 

3.  A   barrel   of    weight,    w,    rests   upon   a   floor   in   stable 
equilibrium,   because  its  centre  of  gravity   is  at  a  distance,   d, 
below  the  axis  :  find  the  work,   W,  necessary  to  start  the  barrel 
rolling.     Prob.      Q. 

4.*  State  some  of  the  most  important  properties  of  the  centre 
of  gravity.     Q.  XXII.-XXVII. 

5.  Show  that  work  must  be  done  upon  a  body  to  displace  it 
from  a  position  of  stable  equilibrium.     D.  50. 

6.  Show  that  work  is  done  by  a  body  when  it  is  displaced 
from  a  position  of  unstable  equilibrium.     D.  50. 

7.  Show  that  no  work  is  done  upon  or  by  a  body  when  it  is 
displaced  from  a  position  of  neutral  equilibrium.     D.  50. 

8.  State  the  criterion  of  the  stability  of  equilibrium  in  terms 
of  the  total  work  done  upon  or  by  a  body  when  displaced  from 
its  position  of  equilibrium.     D.  50. 

9«f  What  condition  of  equilibrium  is  assumed  to  exist  in  an 
ideal  lever  ? 

10. t  State,  in   general,   what   kind  of  equilibrium  exists   in 
various  ideal  machines  or  "  mechanical  powers." 


XXIX. 

PRINCIPLE    OF    WORK    APPLIED    TO    MECHANICAL 

POWERS. 


,  by  geometry,  that  the  work  spent  by  a  force  acting 
upon  an  ideal  lever  is  equal  to  the  work  utilized  in  the  weight 
raised. 

.    2.*  State  the  "  principle  of  work  "  (or  "  virtual  velocities  ") 
as  applied  to  the  lever.     D.  49. 

3.*  What  assumptions  (in  regard   to  loss  of  work  by  friction, 
etc.)  are  made  in  the  investigation  of  ideal  machines?      (D.  49). 

4.  What  distinction   is  made  between   the  force  known   as 
the  "power"  and  the  "weight"  in   various  machines?     D.  59. 

5.  What  is  meant  by  mechanical  advantage  ?     D.  59. 

6.  Name  some  of  the  most  important  "  mechanical  powers." 
D.  55- 

7.  Show  that  the  wheel  and  axle  may  be  regarded  as  an 
endless  lever.     D.  60. 

8.  Find  (by  the  principle  of  work)  the  relation  between  the 
power,  weight,  and  radii  of  a  wheel  and  axle.     Prob. 

9.  How  would  the  results  obtainable  with  a  wheel  and  axle 
be  modified  by  the  thickness  of  the  ropes,  if  these  were  perfectly 
flexible  ?     D.  60. 

ro.f  In  what  way  would  the  stiffness  of  the  ropes  of  a  wheel 
and  axle  modify  the  result  if  the  ropes  were  perfectly  elastic,  so 
that  no  work  need  be  spent  in  bending  them  ? 

ii.  Show  that  a  pulley  may  be  regarded  as  an  endless  lever. 
D.  61. 

i2.f  What  assumption  as  to  the  tension  of  the  cords  passing 
round  one  or  more  pulley-wheels  is  made  in  calculating  the 
mechanical  advantage  of  ideal  pulleys  ? 


43  MECHANICAL  POWERS.  [XXIX. 

13.*  What  mechanical  advantage  is  gained  by  a  single  fixed 
pulley  ?  D.  61. 

i4.f  What  allowance  must  be  made  for  the  weight  of  a 
movable  pulley  ?  for  the  weight  of  the  cords  ? 

154  How  is  the  influence  of  the  weight  of  pulleys  and  cords 
eliminated  in  experimental  demonstrations  ? 

1 6.*  State  the  mechanical  advantage  of  a  single  movable 
pulley,  to  which  the  weight  is  attached,  assuming  the  cords 
parallel.  D.  61. 

17.*  State  in  general  the  mechanical  advantage  of  a  pulley 
niOv'ed  by  x  cords  with  uniform  tension  in  the  same  direction. 
Prob.  D.  62. 

1 8.*  Show  that  the  principle  of  work  applies  to  the  case  of  a 
pulley  with  x  cords,  as  in  the  last  question.  Prob.  D.  61. 

19.  Find  the  mechanical  advantage  of  a  series  of  x  single 
pulleys,  each  attached  to  the  power  cord  of  the  next.  Prob. 
D.  63. 

20. f  What  is  meant  by  an  ideal  smooth  surface  or  plane?  and 
what  limitations  exist  in  the  direction  of  the  force  which  such  a 
surface  is  capable  of  exerting  upon  a  body  ? 

21%  Describe  one  or  more  experiments  with  inclined  planes 
approximating  to  the  ideal  conditions. 

22.  Explain   the  resolution  of  forces   in  a   body  resting  on  a 
perfectly  smooth  inclined  plane.     D.  64. 

23.  A  body  of  weight,  w,  rests  on  an  inclined  plane  of  length, 
AC,  base,  AB,  and  height,  BC:  find  the  force  by  which  it  is  held 
in    place,    (i)    if   this   force    is    tangential,    (2)    if   the   force   is 
horizontal.     Prob.     Q. 

24.  Show  that  the  principle   of  work   applies  to  the  forces 
known  as  the  "  power  "  and  the  "  weight  "  in  an  inclined  plane. 
D.  65,  66. 

25.  Show  that  a  wedge  may  be  regarded  as  a  special  case  of 
inclined  plane.     D.  67. 


44  MECHANICAL  POWERS.  [XXIX. 

26.  Find  the  mechanical  advantage  of  an  ideally  smooth 
wedge  of  length,  /,  and  thickness,  /.  Prob.  Q. 

27.*  Show  that  a  screw  can  be  treated  as  a  special  case  of 
inclined  plane.  D.  68. 

28.*  Find  the  mechanical  advantage  of  a  screw  of  circum- 
ference, c,  and  distance,  d,  between  threads  measured  parallel  to 
the  axis  of  the  screw,  if  the  force  is  applied  at  the  circumference, 
at  right-angles  to  the  axis.  Prob.  Q. 

29.*  By  what  means  is  the  mechanical  advantage  of  a  screw 
greatly  increased  in  ordinary  screw  presses  ?  D.  69. 

30.*  Find  the  mechanical  advantage  of  a  screw  press  with 
arms  of  length,  /,  and  threads,  at  the  distance,  d.  D.  69. 

3i.*fKstimate,  approximately,  the  force  which  can  be  brought 
to  bear  in  an  ordinary  screw  press,  neglecting  friction. 

32.*  Show  that  the  principle  of  work  applies  to  the  screw 
press. 

33.  Show  that  all  the  mechanical  powers  are  reducible  to  two 
types  (the  lever  and  the  inclined  plane). 

34.*  State,  in  general,  the  application  of  the  principle  of 
work  to  mechanical  powers.  D.  49. 

35.*  What  is  meant  by  the  statement  that  "  what  is  gained  in 
power  is  lost  in  speed  ?"  Prob.  Q. 

36.*  An  ideal  machine  is  actuated  by  a  force,  /%  acting  through 
a  distance,  D,  and  produces  a  force,/",  acting  through  a  distance, 
d  ;  express  each  of  the  four  quantities,  /%  /"  D,  and  d,  in  terms 
of  the  other  three. 

37.*  In  the  last  question,  state  what  must  be,  and  what  need 
not  be  known  about  the  construction  of  the  machine,  in  order 
that  the  principle  of  work  may  be  applied.  Q. 


45 


XXX. 

CONSERVATION  OF  WORK. 

i.f  A  weight,  w,  is  moved  through  the  distances,  AB,  BC, 
.  .  .  YZ,  all  lying  in  the  same  vertical  line.  Show  that  the  work 
done  is  the  same  (w  X  AZ},  whatever  may  be  the  position  of 
the  intermediate  points,  B,  C,  etc. 

2.  Prove  that  if  a  weight,  w,   is  moved  along  any  broken 
path,  A"  B"  C"  .   .   .  Z" ,  the  work  done  is  the  same  as  along  the 
projection  of  this  path,  A' B' C'  .   .   .  Z1 ',   on  a  vertical  line,  and 
hence,  equal  to  w  X  A'Z'.     Prob. 

3.  Show  that  in  returning  from  Z"  to  A",  the  weight  gives 
out  the  same  amount  of  work  that  it  received  between  A"  and 
Z".     Prob.     D.  123. 

4.  Show  that  work  done  against  constant  forces,  like  those 
exerted  by   gravity,    depends   only  upon    the    initial    and   final 
positions  of  a  body,  and  is  independent  of  the  path  of  the  body. 
Q.     (D.  123). 

5.  Show  that  when  a  body  returns  by  any  "  closed  path  "  to 
its  original  position,  the  total  work  done  upon   it,    or  by  it,    is 
zero.     Q.     (D.  123). 

6.f  Discuss,  from  the  point  of  view  of  the  last  question,  the 
possibility  of  various  proposed  forms  of  "perpetual  motion." 
(D.  49)- 

7.  State  the  relation  between  the  work  received  by  a  body 
in  one  part  of  a  closed  path,  and  that  given  out  by  it  in  the 
remainder  of  its  path.  Q. 

8.*fWhat  is  meant  by  the  "conservation  of  work"  in 
-mechanics  ? 


XXXI. 

LOSS  OF  WORK  BY  FRICTION. 

i.*  What  element,  entering  into  the  working  of  all  practical 
machines,  causes  a  departure  from  ideal  conditions  ?  D.  49, 
125,  131. 

2.*  Define  (kinetical)  friction  (D.  131),  and  state  the 
peculiarity  of  sign  in  the  work  to  which  it  gives  rise. 

3-t  Describe  and  explain  one  or  more  experiments  showing  a 
great  discrepancy  between  the  force  required  to  start  a  body 
sliding  (no  matter  how  slowly)  over  a  surface,  and  that  required 
to  maintain  the  motion.  (D.  131,  132). 

4. |  State  the  results  of  one  or  more  experiments  showing 
whether  the  force  required  to  drag  a  body  over  a  surface  is  or  is 
.not  affected  by  the  velocity.  (D.  131). 

1    5.f  Why  is  it  necessary,  in  measuring  forces  due  to  friction, 
that  bodies  should  be  moved  with  uniform  velocity  ? 

6. 1  State  the  results  of  one  or  more  experiments  showing  the 
relation  between  the  force  required  to  drag  a  body  over  a  surface, 
and  the  (normal)  force  urging  the  body  and  the  surface  together. 
D.  131. 

7. J  What  is  the  result  of  varying  the  ?rea  of  rubbing 
surfaces,  without  varying  either  their  nature  or  the  force  urging 
them  together  ? 

8.     What  is  meant  by  a  ' '  coefficient  of  friction  ?"   D.  131,  132. 

9-J  Describe  one  or  more  experiments  in  which  the  coefficient 
of  friction  between  two  horizontal  surfaces  is  determined. 

10.  A  body  of  weight,  W,  requires  a  force,  w,  to  drag  it  with 
uniform  velocity,  along  a  horizontal  surface  ;  find  the  coefficient 
of  friction  between  the  body  and  the  surface.  Prob.  Q. 


47  LOSS    OF  WORK    BY    FRICTION.  [XXXI. 

ii.  What  is  meant  by  the  "limiting  angle  of  friction  "  for 
two  surf  aces?  D.  132. 

1 2. J  Explain  the  experimental  determination  of  coefficents  of 
friction  by  means  of  an  inclined  plane.  D.  133. 

13.  A  body  slides  with  uniform  velocity  down  an  inclined 
plane  with  base,  AB,  and  height,  BC:  find  the  coefficient  of 
friction.  D.  133. 

14. f  Give  some  idea  of  the  magnitude  of  coefficients  of  friction 
between  wooden  or  metallic  surfaces,  and  the  effect  of  grease  or 
oil  upon  these  coefficients. 

15.  t  Show  that  the  mechanical  advantage  of  a  wedge  cr  screw, 
no  matter  how  fine  the  pitch,  cannot  exceed  the  reciprocal  of  the 
coefficient  of  friction. 

i6.f  Find  a  relation  between  the  weight,  W,  of  a  truck,  the 
diameters,  D,  and  d,  of  the  wheel  and  axle,  the  coefficient  of 
friction,/,  of  the  (loose)  bearing  upon  the  axle,  and  the  force,  F, 
required  to  pull  the  truck.  Illustrate  by  a  numerical  example. 

ry.f  Find  the  difference  between  the  forces,  F'  and  F" ,  trans- 
mitted by  parallel  cords  to  and  from  a  pulley  of  diameter,  Z?, 
with  axle  of  diameter,  d,  and  coefficient  of  friction,  /,  on  its 
loose  bearing,  (calling  the  force  borne  by  the  pulley  F'  -\-  F"). 

i8.f  Trace  out  the  effect  in  a  double  block  (with  forr  cords) 
of  a  loss  of  10  per  cent.,  20  per  cent.,  etc.,  in  the  tension  of  each 
cord  passing  round  each  wheel,  on  the  force  required  to  lift  a 
given  weight  by  the  block,  remembering  that  the  weight  raised 
is  given  by  the  sum  of  the  tensions  on  the  cords. 

ig.f  Find,  conversely,  the  weight  on  the  block  necessary  to 
overcome  a  given  force  at  one  end  of  the  cord. 

20.  f  Explain  the  enormous  loss  of  work  in  long  trains  of 
clock-work,  and  the  surprising  effects  of  oil  upon  these. 


XXXII. 
EFFICIENCY. 

i. J  Describe  one  or  more  experiments  showing  that  the  work 
spent  upon  the  cord  of  a  tackle  in  raising  a  weight  is  greater 
than  that  utilized  through  the  pull  upon  this  cord  exerted  by  the 
descending  weight. 

2.t  What  name  is  given  to  the  ratio  of  the  work  spent  to  the 
work  utilized  in  a  given  case  ? 

3.     Define  "efficiency,"  as  used  in  mechanics.     Q.  2. 

4.f  Distinguish  between  the  efficiency  of  a  tackle  as  a 
machine  for  raising  weights,  its  efficiency  as  a  machine  for 
utilizing  work  done  by  the  descent  of  a  weight,  and  its  efficiency 
as  a  machine  for  storing  useful  work. 

5.f  State  a  certain  necessary  relation  between  the  three 
efficiencies  in  the  last  question. 

6.  A  man  slides  a  box  up  into  a  wagon  along  a  board  inclined 
at  the  limiting  angle  of  friction  :   (Q.  XXXI.)  find  the  efficiency 
of  his  combination.     Prob. 

7.  Show  that  the  efficiency  of  a  screw,  irreversible  through 
friction,  cannot  exceed  50  per  cent.     Prob. 

8.f  Show,  from  a  point  of  view  of  efficiency,  the  disadvantage 
of  .a  screw  with  too  fine  a  thread  (or  of  a  differential  screw). 


49 


XXXIII. 
POWER. 

i. J  Describe  one  or  more  experiments  with  an  ergometer  (or 
friction-brake),  showing  how  it  is  possible  to  measure  the  work 
spent  by  a  given  agent  in  a  given  time. 

2.J    Describe  a  "transmission  dynamometer,"  and  its  uses. 

3.f  What  name  is  given  to  the  quotient  of  work  by  time?  to 
the  product  of  force  and  velocity  ? 

4.f  Distinguish  power,  in  its  technical  sense,  from  power  as  a 
name  for  one  of  the  forces  in  a  machine. 

5«f  Define  powrer  in  two  different  ways,  and  show  that  these 
definitions  are  identical.  Q.  3. 

6.f  Define  "horse-power"  (English  or  French),  the  erg  per 
second,  and  the  "watt." 

7-t  Show  that  the  horse-power  varies  slightly  in  different 
latitudes,  but  that  the  erg  and  the  watt  are  constant. 

8.f  Find  the  horse-power  of  a  locomotive  with  two  cylinders 
100  square  inches  in  section,  and  double  two-foot  stroke,  making 
125  revolutions  per  minute  under  a  mean  pressure  of  33  Ibs.  of 
steam  per  square  inch,  in  both  the  forward  and  backward  stroke, 
making  no  allowance  for  friction. 

g.f  Find  the  power,  P,  spent  upon  a  water-motor  with  piston 
of  area,  a  (sq.  cm.),  making  n  double  strokes  of  length,  /,  (cm.) 
in  every  second,  under  a  pressure,  p  (dynes  per  sq.  cm.),  and 
state  in  what  units  the  result  is  expressed. 

io.f  Show  that  the  power,  P,  spent  upon  a  motor  (by  an 
incompressible  liquid)  is  equal  to  the  product  of  the  pressure,  p, 
per  unit  of  area,  and  current,  c,  in  units  of  volume  per  unit  of 
time. 

ii.t  Show  that  (as  a  consequence  of  the  principle  of  the  last 
question),  if  pressure  is  transmitted  by  an  incompressible  fluid 
without  loss,  power  must  also  be  transmitted  without  loss. 


XXXIV. 

SOLIDS  AND  FLUIDS,  DISTINGUISHED. 

1. 1  Describe  an  experiment  (with  OERSTED'S  piezometer), 
showing  that  liquids  are  compressible.  D.  130. 

2.*  Distinguish  fluids  from  solids  (in  respect  to  their  relative 
resistance  to,  and  limits  of  recovery  from,  changes  of  form  or 
shape  unaccompanied  by  changes  of  bulk  or  volume).  (D.  129). 

3.f  Distinguish  fluids  from  solids  in  respect  to  their  rates  of 
yielding  to  tangential  or  transverse  forces,  and  state  whether  this 
distinction  is  one  of  kind  or  of  degree. 

4.f  What  name  is  given  to  that  property  in  solids  which  is 
connected  with  their  slow  yielding  under  transverse  stresses  ? 

5.f  When  is  a  solid,  and  when  is  a  fluid  said  to  be  especially 
viscous?  (D.  135). 

6.f  Show  that  viscosity  is  used  in  two  opposite  senses  in  its 
applications  to  solids  and  fluids. 

y.f  Distinguish  frictional  forces  within  the  body  of  a  fluid 
(whether  due  to  viscosity  or  not)  from  forces  due  to  the  rubbing 
of  two  solid  surfaces,  in  respect  to  their  dependence  upon  (i)  the 
compression,  (2)  the  area,  and  (3)  the  relative  velocity  of  the 
moving  parts. 

8.*  State  some  of  the  characteristic  distinctions  between 
solids  and  fluids.  Q. 


XXXV. 

LIQUIDS  AND  GASES  DISTINGUISHED. 

1. 1  Describe  an  experiment  (with  a  gas-bag  and  air-pump) 
illustrating  the  indefinite  expansibility  of  a  gas.  D.  194. 

2.*  Distinguish  fluids  into  two  classes,  according  to  their 
tendency  toward  uniform  distribution  throughout  any  space 
within  which  they  are  confined.  D.  194. 

3.*  How  do  liquids  compare,  as  a  class,  with  gases,  in  respect 
to  density  ? 

4.*  How  do  liquids  compare,  as  a  class,  with  gases,  in  respect 
to  compressibility  ? 

5.f  What  is  meant  by  the  "  free  surface  "  of  a  fluid,  and  what 
kind  of  fluids  alone  present  such  surfaces  under  ordinary 
conditions  ? 

6.J  Describe  one  or  more  experiments  (e.g.,  with  floating 
needles),  showing  that  the  surfaces  of  liquids  resist  deformation 
to  a  slight  extent.  D.  159. 

7.J  Describe  one  or  more  experiments  (with  camphor  and 
water,  oil  films  on  water,  and  the  effects  of  local  heat  or  alcohol) 
showing  that  slight  tangential  forces  are  exerted  by  the  surfaces 
of  liquids.  D.  192. 

8.  J   Describe  one  or  more  experiments  (with  PLATEAU'S  films) 
showing  that  the  surface  of  a  film  is  a  minimum  consistent  with 
its  boundary.     (D.  186). 

9.  Explain  the  spherical  shape  of   rain-drops   or   bubbles. 
D.  189. 

10. J  Describe  one  or  more  experiments  showing  that  films  of 
liquid,  when  free  to  do  so,  cause  their  boundaries  to  contract. 

ii.  State  the  influence  of  the  length,  the  breadth,  and  the 
thickness  of  a  film  on  the  force  with  which  it  contracts 
longitudinally.  D.  185. 


52  LIQUIDS  AND    GASES   DISTINGUISHED.  [XXXV. 

12.  What  is  meant  by  ".surface  tension  ?"     D.  185. 

13.  How  many  surfaces  is  a  (soap-bubble)  film  considered  to 
possess  ?  and  how  is  the  surface  tension  of  such  a  film  measured  ? 
D.  188. 

14.*  Describe  one  or  more  experiments  illustrating  adhesion 
between  a  liquid  and  a  solid  which  it  wets. 

15.*  Describe  the  surfaces  of  liquids  of  two  different  kinds, 
near  the  edges  of  the  vessels  containing  them.     D.  182. 

1 6.*  What  is  meant   by  a  capillary  tube  ?  by  capillary  forces? 
and  by  capillarity  in  general  ? 

17.*  Describe  the  rise  and  fall  of  liquids  caused   by  capillary 
tubes.     D.  182. 

1 8.*  When  will  a  liquid  rise  in  a  capillary  tube,  and  when  will 
it  be  depressed  by  it  ?     D.  182. 

19.  Describe  the  "  meniscus  "  (or  curved  surface)  of  a  liquid 
accompanying   (i)  capillary  ascensions  and  (2)  capillary  depres- 
sions.    D.  182,  190. 

20.  Kxplain  the  rise  of  liquids  in  capillary  tubes  which  they 
wet,  and  their  depression  by  capillary  tubes  which  they  do  not 
wet.     D.  1 86. 

21.  State  some  of  the  conditions  which  influence  the  amount 
of  the  rise  or  fall  of  a  liquid  in  a  capillary  tube.     D.  183,  184. 

22.f  What    is    meant    by    the   height   of  the   meniscus    (or 
sagitta)  ? 

23.  What  is  meant  by  the  angle  of  contact  between  a  liquid 
and  a  solid?     D.  184,  185. 

24.  Find  the  height,   /i,  of  a  column  of  liquid  of  density,  d, 
sustained  by  a  surface  tension  (in  gravitation  units)   t,  in  a  tube 
of  radius,  r,  assuming  that  the  tube  is  wet  by  a  film  of  liquid 
tangent  to  its  surface.     D.  186. 

25.  State  the  law  of  diameters  governing  capillary  ascensions 
and  depressions.     D.  184. 

26. J   Describe  one  or  more  experiments  (e.g.,   with    inclined 
plates)  illustrating  the  law  of  diameters. 


53  LIQUIDS  AND    GASES   DISTINGUISHED.  [XXXV. 

27.  f  How  high  are  liquids  known  to  rise  by  capillary  action  in 
certain  vegetable  structures  ? 

28. f  Give  reasons  for  supposing  that  a  practical  limit  exists  in 
the  height  attainable  by  capillary  action. 

29.  J  Describe  an  experiment  with  an  air-pump,  showing  that 
capillary  ascensions  and  depressions  are  not  due  to  atmospheric 
pressure.  (D.  185). 

30.J Describe  one  or  more  experiments  illustrating  the  tensile 
strength  under  certain  specified  conditions,  of  columns  of  liquid 
of  considerable  cross  section. 

31.*  In  what  class  of  fluids,  alone,  are  phenomena  due  to 
capillarity,  cohesion,  or  surface  tension  perceptible  ?  Q. 

32.*  State  some  of  the  characteristic  distinctions  between 
liquids  and  gases.  Q. 

33. 1  Describe  one  or  more  experiments  illustrating  effects  of 
endosmose  (or  diffusion  through  a  diaphragm)  and  endosmotic 
pressure.  D.  193. 

34.  Distinguish  solutions  into  two  classes  with  respect  to  the 
facility  with  which  they  pass  through  a  diaphragm.     D.  193. 

35.  Describe  the  process  of  separating  colloids  and  crystalloids 
by  "dialysis."     D.  193. 

36.  Describe  one  or  more  experiments  illustrating  the  diffusion 
of  gases,  and  the  pressures  to  which  such  diffusion  may  give 
rise.     D.  193,  227. 


54 


XXXVI. 
FUNDAMENTAL   ASSUMPTIONS    IN    HYDROSTATICS. 

i.f  What  assumptions  are  usually  made  in  the  solution  of 
problems  in  hydrostatics,  with  respect  to  viscosity,  capillarity, 
osmosis,  etc.  ? 

2.  Why  does  not  viscosity  enter  into  problems  is  hydrostatics? 
D.  135- 

3-t  State  reasons  for  assuming  that  capillary  forces  may  be 
neglected  in  the  treatment  of  bodies  of  liquid  of  considerable 
size,  or  in  the  case  of  solids  of  considerable  size  immersed  in 
these  liquids. 

4.J  Give  experimental  grounds  for  the  assumption  that  forces 
or  pressures  due  to  osmosis  or  to  diffusion  are  not  perceptible 
in  the  wide  channels  usually  employed  in  hydrostatics. 

5-t  Show  that,  in  the  absence  of  viscosity  and  capillarity, 
normal  forces  and  pressures  are  the  only  ones  which  need  be 
taken  into  account.  (D.  135). 

6.  Distinguish    hydrostatics    from    hydrokinetics,    treating 
both  as  branches  of  hydrodynamics.     D.  134. 

7.  Distinguish  pneumatics  from  hydrostatics.     D.  3. 

8.f  To  what  extent  are  the  properties  of  gases  studied  under 
the  head  of  hydrostatics  ? 

Q.f  State  certain  obvious  experimental  evidence  of  the  fact 
that  fluids,  when  undisturbed  for  a  long  time,  fall  (practically) 
into  a  state  of  equilibrium. 

io.f  What  general  principle  would  lead  to  the  anticipation  of 
the  result  stated  in  the  last  question  ? 

u.f     What  hypothesis  (as  to  the  equilibrium  of  fluids)  lies  at 
the  foundation  of  hydrostatics  ? 


55 


XXXVII. 
CONDITIONS  OF  EQUILIBRIUM  IN  FLUIDS. 

i.f  Show  (by  considerations  relating  to  the  centre  of  gravity) 
that  a  bubble,  contained  in  a  liquid  within  any  enclosure,  seeks 
the  highest  possible  point. 

2.  Describe   the  construction   and  use  of  an  ordinary  spirit- 
level,  its  attachment  to  telescopes  with  cross-hairs,  the  conditions 
of,  and  one  or  more  processes  of  testing  its  sensitiveness,  and  the 
method  of  eliminating  errors  of  adjustment.     D.  180,  181. 

3.  State  the  condition  of  equilibrium  in  the  free  surface  of  a 
liquid.     D.  140. 

4.  Show,  by  the  resolution  of  forces,  that  if  the  free  surface 
of  a  fluid  is  not  horizontal,  it  is  not  in  equilibrium.     D.  140. 

5.f  Show,  by  considerations  relating  to  the  centre  of  gravity, 
that  the  bounding  surface  between  any  two  fluids  of  unequal 
density  is  in  equilibrium  only  when  horizontal.  (D.  145). 

6. 1  Describe  one  or  more  experiments  illustrating  the  equili- 
brium of  bounding  surfaces  between  different  fluids.  D.  145. 

7.  Show  that  the  equilibrium  of  a  fluid  would  not  be 
disturbed  if  any  portion  of  it  should  become  rigid,  or  should  be 
replaced  by  a  fixed  rigid  body.  D.  153. 

8.f  Show  that  it  is  possible,  by  imagining  rigid  supports 
substituted  for  certain  portions  of  a  body  of  fluid,  to  treat  the 
remainder  as  if  contained  in  a  £/-tube,  or  in  communicating 
vessels  of  any  shape,  without  any  change  of  equilibrium. 

9.*  State  the  condition  of  equilibrium  of  a  liquid  contained  in 
communicating  vessels.  D.  178. 

i o.*J  Describe  one  or  more  experiments  illustrating  the  fact 
that  a  liquid  stands  at  the  same  level  in  communicating  vessels. 
D.  178. 


56  CONDITIONS    OF   EQUILIBRIUM    IN    FLUIDS.         [xXXVII. 

ii.  Explain  the  construction  and  use  of  a  water-level. 
D.  179. 

i2.f  Show  (by  the  principle  of  action  and  reaction)  that  the 
resultant  of  all  the  forces  exerted  by  a  fluid  upon  the  walls  of  its 
enclosure  must  be  equal  and  opposite  to  the  resultant  of  the  forces 
exerted  by  the  walls  of  the  enclosure  upon  the  fluid,  and  hence 
(by  the  fundamental  principles  of  equilibrium)  equal  to  the 
weight  of  the  fluid.  (D.  148). 

i3-t  Show  that  the  weight  of  any  portion  of  a  fluid,  of 
whatever  shape,  is  equal  and  opposite  to  the  resultant  of  the 
forces  exerted  upon  this  portion  by  the  body  of  fluid  which 
surrounds  it. 

14.  Show  that  the  difference  between  the  forces  exerted  upon 
the  top  and  bottom  of  a  column  of  fluid  with  vertical  sides  (from 
which  the  column  is  supposed  to  derive  no  support)  must  be  equal 
to  the  weight  of  the  column.     D.  139. 

15.  Show  that  the  resultant  forces  upon  the  two  ends  of  a 
fluid  prism  with  horizontal   axis  must  be  equal    and  opposite. 
D.  138. 

1 6.  Find  by  the  triangle  of  forces,   the  relation  between  the 
resultant  forces  upon  the  three  rectangular  faces  of  a  triangular 
fluid  prism  of  negligible  weight,  and  show  (by  similar  triangles) 
that  the  three  forces  are  proportional  to  the  areas  of  the  three 
faces  in  question.     D.  137. 


57 


XXXVIII. 
PRESSURE. 

i.f  What  is  meant  by  intensity  of  pressure?  and  in  what 
units  is  it  expressed  ?  D.  136. 

2.f  In  what  sense  is  the  word  "  pressure,"  when  unqualified, 
used  by  most  modern  writers  ?  and  how7  does  this  differ  from 
another  sense  in  w7hich  it  was  sometimes  employed  by  earlier 
writers?  D.  136. 

3-f  Find  the  (intensity  of)  pressure,  p,  (expressed  in  gravi- 
tation units)  exerted  by  a  weight,  w,  upon  a  horizontal  surface 
of  area,  a. 

4.  A  weight,  w,  of  fluid  is  contained  in  a  tube  with  vertical 
sides,    and  with   an   area  of  cross-section,  a.     Show  that  if  the 
upper  surface  is  free  from  force,  the  weight  of  the  fluid   is  borne 
by  the  bottom  of  the  tube,  alone  ;  and  find  the  pressure  due  to  the 
weight  of  the  fluid  upon  the  bottom  of  the  tube,  supposing  it  to 
be  horizontal.      Prob.     Q.  XXXVII.-XXXVIII. 

5.  Find  the  weight,  w,  of  fluid  of  density,  d,  standing  at  a 
height,  h,  in   a  tube  with  vertical   sides  and   horizontal  base  of 
area,  a  ;   find  also  the  pressure   upon   the   bottom   of   the  tube. 
Prob.     Q.  XXXVII.-XXXVIII. 

6.  Find,    as  in  the   last  question,   the   pressure,  p,    at    the 
bottom  of  a  vertical  column  of  fluid  of  density,  d,  and  depth,  h, 
when  in  equilibrium  with  the  surrounding  fluid.     Prob.     Q. 

7.  •   Show  (by  the  principle  of  action  and  reaction)  that  the 
upward  pressure  beneath  any  horizontal  area,  a,  in  the  body  of  a 
fluid,  must  be  equal   to  the  downward  pressure  on  this  area,  due 
to  the  fluid  above  it.     Prob. 

8. \  Describe  one  more  experiments  showing  that  the  upward 
pressure  at  a  given  depth  in  a  liquid  is  equal  to  the  downward 
pressure  due  to  a  column  of  liquid  of  the  same  depth  and  density. 
D.  144. 


58  PRESSURE.  [XXXVIII. 

9.  Prove  (by  the  conditions  of  equilibrium  of  the  pressures 
on  the  faces  of  a  small  triangular  prism,  with  axis  horizontal) 
that  the  pressure  upon  any  surface  of  given  area  at  a  given  depth 
in  a  fluid  is  the  same  as  upon  a  horizontal  surface  of  the  same 
area,  and  at  the  same  depth.  D.  137.  Q.  XXXVII.  15. 

10.*  Explain  the  statement  that  the  pressure  of  a  fluid  is  the 
same  in  all  directions.  D.  137. 

n.  Prove  (by  considering  the  conditions  of  equilibrium  of 
the  forces  on  the  ends  of  a  prism  with  horizontal  axis)  that  the 
pressure  at  two  points  at  the  same  level  in  a  fluid  must  be  the 
same.  D.  138. 

12.  Prove  (by  considering  the  conditions  of  equilibrium  of  a 
column  of  fluid  with  vertical  sides)  that  in  descending  through 
the  vertical  distance,  h,  in  a  liquid  of  density,  d,  the  pressure 
(measured  in  gravitation  units)  increases  by  the  amount,  hd, 
whether  the  starting  point  is  in  the  free  surface  of  the  liquid  or 
not.  D.  139. 

I3.|  Describe  one  or  more  experiments  (with  communicating 
tubes)  illustrating  the  equality  of  pressure  between  columns  of 
liquid  of  given  vertical  height  in  tubes  having  different 
inclinations. 

14.  Show  (by  a  zigzag  of  horizontal  and  vertical  prisms)  that 
the  pressure   at  two  points  on  the  same  le\el  in  communicating 
vessels  must  be  the  same.     Prob.     D.  139,  147. 

15.  Show  (as  in  the  last  question)  that  the  pressure  of  a  fluid 
of  density,  d,  at  a  depth,  7z,  below  a  given  surface  is  greater  than 
at  this  surface  by  the  amount,  hd,  regardless  of  the  shape  of  the 
vessel.     D.  139,  147. 

i6.*J Describe  one  or  more  experiments  illustrating  the  fact 
that  the  force  exerted  on  a  given  area  by  a  liquid  of  given  depth 
and  density  is  independent  of  the  shape  of  the  vessel  containing 
the  liquid.  D.  147. 

I7.J  Describe  one  or  more  experiments  {e.g. ,  with  E.  H. 
HAUL'S  pressure-gauge)  illustrating  the  fact  that  the  pressure 


59  PRESSURE.  [XXXVIII. 

in  a  liquid  increases  with  the  depth,   and  is  the   same   in  all 
directions. 

1 8.  Discuss  the  applicability  of  the  principles  of  hydrostatic 
pressure  to  the  case  of  liquids  in  capillary  tubes.     D.  190. 

19.  Show  that  the  pressure  of  a  liquid  just  within   its  curved 
surface  in  a  capillary  tube  must  differ  considerably  from  that  just 
outside  of   it,   and  show  that  the  difference  is  explainable  by 
surface  tension.     D.  190,  191. 

20.  Find  the  sign  of  the  pressure  of  a  liquid  raised  in  vacuo  by 
a  capillary  tube,  on  the  assumption  that  its  pressure  outside  of 
this  tube,  is  zero.     D.  190. 

21.  Explain  the  attraction  between  two  parallel    plates,  or 
between  two  floating  bodies  of  which   (i)  both  are  wet,  or  (2) 
neither  is  wet  by  the  liquid.     D.  190. 


6o 


XXXIX. 
BALANCING  COLUMNS. 

i. *J  Describe  an  experiment  in  which  water  is  introduced  into 
one  branch  of  a  £/-tube  containing  some  mercury.  (D.  146). 

2.*  If  the  water  in  the  last  question  reaches  from  the  level,  a, 
to  the  level,  b,  and  the  mercury  reaches  from  the  level,  b,  down 
to  the  bottom  of  the  £/-tube  and  up  again  to  the  level,  <r,  what 
is  the  function  of  the  mercury  below  b  ?  and  what  is  the  relation 
between  the  pressures  exerted  upon  this  portion  of  the  mercury 
by  the  column  of  water  (ab)  above  it  on  one  side,  and  by  the 
column  of  mercury  (be)  above  it  on  the  other  side?  D.  146. 

3.*fShow  that  the  depth  of  a  £/-tube,  below  the  level  where 
two  liquids  abut,  has  nothing  to  do  with  their  equilibrium. 

4.*t  State  what  portions  of  two  liquid  columns,  abutting  in 
a  £/-tube,  are  considered  as  producing  equal  pressures. 

5.f  Show  that  the  condition  of  equilibrium  between  two 
balancing  columns  is  not  affected  by  atmospheric  pressure, 
provided  that  it  acts  upon  both  alike. 

6.J  Explain  the  use  and  function  of  compressed  air  in 
balancing  liquid  columns  which  cannot  be  allowed  to  abut. 

7.  Point  out  the  liquid  columns  exerting  equal  pressures  on 
compressed  air  contained  in  a  tube  shaped  like  a  W.  Prob.  Q. 

S.I  Explain  the  method  of  balancing  columns  of  liquid  by 
means  of  rarefied  air. 

9.  Point  out  the  liquid  columns  exerting  equal  pressures  in 
an  inverted  F-tube.  Prob.  Q. 

10.*  Find  a  relation  between  the  densities,  D  and  d,  and  the 
vertical  heights,  h  and  H,  of  two  balancing  columns,  and 
express  each  of  the  four  quantities  in  terms  of  the  other  three. 
Prob.  Q. 


6 1  BALANCING   COLUMNS.  [XXXIX. 

ii.  Describe  three  methods  of  balancing  columns,  and  explain 
how  the  relative  density  of  two  liquids  can  be  found  by  each.  Q. 

12. f  Show  that  the  pressures  of  two  balancing  columns  of  the 
heights,  //and  h,  are  not,  under  ordinary  circumstances,  exactly 
equal,  but  differ  by  a  small  amount,  equal  to  the  pressure  of  a 
column  of  air  of  the  height,  H —  h. 

i3.f  State  the  magnitude  and  sign  of  the  correction  for  the 
pressure  of  a  column  of  air  in  some  special  case  (e.g.,  when  a 
column  of  water  differs  in  height  by  800  mm.  from  its  balancing 
column,  in  air  800  times  less  dense  than  water). 

14.  t   Describe  a  mercurial  manometer,  and  show  how  it  can  be 
used  for  measuring  differences  of  pressure.     D.  228. 

15.  Find   the   difference  of  pressure    (in   gravitation   units) 
necessary  to  maintain  a  difference  of  level  of  i  cm.,    100  cm., 
etc.,  in  a  manometer  containing  mercury  of  the  density   13.6. 
Prob.     Q. 

1 6.  Describe  a  "  multiple  branch  manometer,"   and  state  its 
advantages  and  disadvantages.     D.  224. 


OF  THE 

TJBIVEHSIT7 


62 


XL. 
ATMOSPHERIC  PRESSURE. 

1. 1  Describe  a  (BOURDON)  metallic  pressure-gauge,  and  the 
process  of  calibrating  such  a  gauge  by  a  mercurial  manometer. 
D.  226  (225). 

2.f  In  stating  that  the  pressure  of  a  liquid  is  proportional  to 
the  depth,  what  assumption  is  made  as  to  the  pressure  at  the 
surface  of  the  liquid  ? 

3.  If  the  pressure  (due  to  atmospheric  or  other  influences)  at 
the  surface  of  a  liquid,  instead  of  being  zero,  is  P,  what 
correction  must  be  made  in  the  last  statement?  D.  141. 

4-t  Explain  how,  in  the  method  of  graduating  a  pressure- 
gauge,  the  correction  in  the  last  question  might  be  everlooked. 

5.J  Describe  one  or  more  experiments  with  pressure-gauges, 
showing  that  pressures  exist  lower  than  that  of  the  atmosphere. 

6.f  State  reasons  why  a  vacuum,  rather  than  the  atmosphere, 
should  be  taken  as  representing  the  zero  of  pressure.  (D.  191  ?). 

y.f  What  modification  is  necessary  to  convert  a  (BOURDON) 
pressure-gauge  into'  an  Aneroid  barometer  ? 

8.J  Describe  the  Aneroid  barometer  as  usually  constructed. 
D.  206. 

9.J  Describe  the  construction  of  a  simple  form  of  mercurial 
barometer  (the  ''Torricellian  experiment  ")  and  state  what  facts 
it  illustrates.  D.  196,  197,  200. 

10.*  Give  some  idea  of  the  average  height  of  the  barometric 
column  at  the  sea  level  in  inches  and  in  centimetres.     D.  197. 

ii.*  How  is  the  atmospheric  pressure  at  a  given  time  and 
place  measured  ?     Prob.     D.  197. 


63  ATMOSPHERIC    PRESSURE.  [XL. 

12.  Find  the  atmospheric  pressure  (in  gravitation  units) 
which  will  sustain  76  cm.  of  mercury  of  the  density,  13.596. 
Prob.  Q. 

i3-t  Define  a  pressure  of  "one  atmosphere"  according  to 
French,  and  more  or  less  general  scientific  usage.  (D.  198). 

14.  Calculate  the  force  on  a  given  surface  (e.g.,  the  human 
body)  due  to  atmospheric  pressure.  Prob.  Q. 

15.^  Describe  one  or  more  experiments  illustrating  effects  of 
atmospheric  pressure  (e.g.,  Magdeburg  hemispheres,  burst 
membranes,  mercury  forced  through  wood,  fountain  in  vacuo, 
etc.).  D.  236. 

16.  Find  the  height  of  a  column  of  air  of  density,  .0012, 
which  will  exert  a  pressure  of  "one  atmosphere"  (76  cm., 
mercury  of  density,  13.6).  Prob.  Q. 

17..  What  is  meant  by  the  "height  of  the  homogeneous 
atmosphere?"  D.  210. 

1 8.  Describe  and  explain  (PASCAL'S)  experiments  in  ascending 
hills  with  a  barometer.     D.  199. 

19.  Find  the  difference    in   pressure    (in  gravitation   units) 
between  two  points  on  a  mountain  differing   10,  100,    1000,  etc., 
metres  in  altitude,  when  the  mean  density  of  the  atmosphere  is 
.0012.     Prob.     Q. 

20.  Apply   the   last    question    to   points   higher   up   on    the 
mountain,  where  the  density  of  the  air  becomes   .0011,   .0010, 
.0009,  .0008,  .0007,  or  even  .0006.     Prob.     Q. 

2 1 .  Explain  the  estimation  of  heights  by  the  barometer,   the 
necessity   of   taking   into  account   the   density   of   the   air,    its 
variations,    if    considerable,    and    the    use    of    tables    in    this 
connection.     D.  209. 

22.  What  relation  usually  exists  between  the  fluctuations  of 
a  barometer  and  meteorological  changes  ?     D.  215. 

23.  What  relation  exists  between  the  height  of  the  mercurial 
barometer  and  that  of  one  containing  water  or  other  liquid  ? 
D.  197. 


64 


XLI. 
FLOW   OF    LIQUIDS    AFFECTED    BY    AIR    PRESSURE. 

i.J  Describe  a  "  pipette,"  (or  tube  closed  by  the  finger  at  one 
end),  and  explain  the  agency  of  the  atmosphere  in  sustaining 
the  weight  of  the  liquid,  and  in  transmitting  it  to  the  place  where 
it  is  felt.  (D.  269). 

2.  J  Describe  one  or  more  experiments  illustrating  the  principle 
of  the  pipette.  D.  269. 

3.|  Describe  and  explain  the  action  of  an  "intermittent 
fountain."  D.  270. 

4.J  Describe  a  "  MARIOTTE'S  bottle,"  and  state  the  conditions 
under  which  it  will  give  rise  to  a  flow  of  water  under  constant 
pressure.  D.  275. 

5.  State   the   uses   of  a    MARIOTTE'S   bottle   for   obtaining 
constant  pressure  or  constant  suction.     D.  275. 

6.  Find  the  pressure  (or  suction)  due  to  a  liquid  of  density,  d, 
issuing  from  a  MARIOTTE'S  bottle,  if  h  is  the  height  of  the  inlet 
above  the  outlet.     Prob.     Q. 

7. 1   Describe  and  explain  the  action  of  a  "siphon."     D.  271. 

8.  What  relation  exists  between   the  height  of  a  barometer 
filled  with  a  given  liquid  and    the  maximum    height    through 
which    that    liquid    can    be    raised    by    an    ordinary    siphon  ? 
Prob.     Q. 

9.  Why  must  air  be  excluded  from  a  siphon  ?  and  through 
what  device   is  this  accomplished   in  small   siphons  ?    in  large 
siphons?     D.  271. 

10.  Explain  the  "vase  of  Tantalus,"  and  state  the  conditions 
of  intermittent  flow  in  terms  of  the  rates  of  flow  into  and  out  of 
the  vase.     D.  274. 

11.  Give    a    possible    explanation    of    certain    intermittent 
springs.     D.  275. 

12%   Describe  and  explain  the  action   of  a  SPRENGEL  pump. 
D.  240. 


XLII. 

PUMPS. 

1.  What  connection  exists  between  the  height  of  a  barometer 
filled  with  a  given  liquid  and  the  maximum  height  to  which  the 
liquid  can  be  raised  by  a  suction  pump?     D.  255. 

2.  Find  the  maximum  height  to  which  water  (of  density,  i) 
^ean  be  sucked  under  atmospheric  pressure   (1033  grams  to  the 
sq.  cm.).     Prob.     Q. 

3.J  Draw  a  diagram  of  an  ordinary  suction-pump  (showing 
the  barrel,  piston,  and  two  valves).  D.  254. 

4.*  Name  the  essential  parts  of  a  suction-pump,  and  explain 
its  action.  D.  254. 

5-f  Show  that  an  ordinary  suction-pump  is  identical  in 
principle  with  one  of  the  simpler  forms  of  air-pump,  and  that  it 
is  use$  as  an  air-pump  in  the  first  stages  of  raising  water.  Prob. 
D.  229,  254. 

6.*f  State  the  use  of  wetting  the  valves  of  a  pump  in  the 
first  stages  of  raising  water. 

y.f  What  modifications  are  necessary  to  convert  a  suction - 
pump  into  a  force-pump  ?  D.  258,  259. 

8.  What  is  meant  by  a  "  plunger,"  and  in  what  respect  does 
it  differ  from  an  ordinary  piston  ?  D.  259. 

9.^  Draw  a  diagram  of  an  ordinary  force-pump,  and  state  its 
advantages  over  a  suction-pump. 

10.     Show  that   a   force-pump   should    not   be   inferior   to   a 
suction-pump,  even  when  suction  alone   is  required.     Prob.     Q. 

n.f  Show  that  a  force-pump  can  be  used  either  for  exhausting 
or  for  compressing  air,  or  other  gases.      (D.  243). 

1 2. |   Draw  a  diagram  of  an   air-pump  with  three  valves,   and 
state  its  practical  advantages.     (D.  235). 


66  PUMPS.  [xui. 

i3-t  Show  that  the  two  sides  of  a  piston  of  any  pump,  raising 
a  liquid  of  density,  d,  through  the  vertical  height,  h,  between 
two  levels  (the  atmospheric  pressure  being  practically  the  same 
at  each  level)  are  subject  to  a  difference  of  pressure,  hd. 

i4.f  Find  the  force  due  to  difference  of  pressure  on  the  two 
sides  of  the  piston  in  the  last  question,  if  its  area  is  a,  also  the 
work,  w,  necessary  to  move  it  against  this  force  (neglecting 
friction)  through  the  distance,  s.  (D.  256). 

15.  Find  the  volume,  v,  and  weight,  w,  of  liquid  delivered  in 
the  last  question  by  the  stroke,  s,   and  calculate  the  work  utilized 
in  raising  it  to  the  height,  h.     Prob. 

16.  Show  that  the  principle  of  work  applies  either  to  suction 
or  to  force-pumps.     Prob.     Q. 

.17. 1   Explain  the  construction  and  use  of  a  hydraulic  press. 
D.  142,  264. 

18.  Find  the  force,  F,  upon  the  large  piston  of  area,  A,  in  a 
hydraulic  press,  due  to  the  force,  /,  on  the  small  piston  of  area, 
a,  (at  the  same  level  as  the  large  piston).     Prob.     Q. 

19.  Find  the  distance,  d,  through  which  the  large  piston  of 
area,    A,   moves  when   the   small    piston,   of   area,  a,   advances 
through  the  distance,  D,  (assuming  the  liquid  incompressible). 
Prob. 

20.  Show  that  the  principle  of  work  applies  to   the  hydraulic 
press.     D.  143. 

2i.f  What  assumption  (as  to  the  levels  of  the  two  pistons)  is 
made  in  elementary  discussions  of  the  hydraulic  press  ? 

22.f  Show  how,  by  the  use  of  a  counterpoise,  corrections  due 
to  differences  of  level  in  a  hydraulic  press  may  be  eliminated. 

23.  What,  in  general,  is  the  effect  of  increasing  the  pressure 
at  a  given  point  in  a  fluid,  by  the  amount,  P,  upon  the  pressure, 
p,  at  any  other  point?     D.  141. 

24.  What  is  meant  by  transmissibility  of  pressure  in  fluid 
systems?     D.  141. 


XLIII. 
CENTRES  OF  PRESSURE  AND  BUOYANCY. 

1.  How  do  you  find  the  magnitude  of  the  force  exerted  by  a 
liquid  upon  a  plane  surface  of  variable  depth?     D.  150,  151. 

2.  Define  centre  of  pressure.     D.  150,  151. 

3.  How  do   you  find  the  centre  of  pressure  on  any  plane 
surface  of  variable  depth  ?     D.  151. 

4.  Where  is  the  centre  of  pressure  upon  a  dam  ?  and  where 
should  a  single  prop  be  applied  to  resist  this  pressure?     Prob.   Q. 

5.  Where  is  the  point  of  application  of  the  forces  exerted 
by  a  liquid  on  a  body  immersed  in   it  ?  on  a  body  floating  upon 
the  liquid  ?     D.  153. 

6.  Define  centre  of  buoyancy.     D.  153. 

7.  State   the  conditions  of  equilibrium  between  the    forces 
due  to  the  weight  of  a  body  and  the  buoyant  action  of  the  fluid 
in  which  it  is  submerged.     D.  155,  156,  157. 

8.  Show  that  the  conditions  in  the  last  question  do  not  hold 
for  a  body  floating  on  the  surface  of  a  liquid.     D.   157. 

9.  State  the  conditions  of  stable  and  unstable  equilibrium, 
for  bodies  floating  on   a  liquid,   in  terms  of  the  path  of  their 
centres  of  gravity,  due  to  rotation  of  the  bodies,  with  constant 
volume  immersed.     Q.  XXV.   17. 

10.     Explain   the   equilibrium   of   a   boat  with  its  centre   of 
gravity  above  the  centre  of  buoyancy.     D.  157. 

n.f  Prove  that  the  stability  of  a  boat  is  always  increased  by 
lowering  the  centre  of  gravity.     (D.  158). 

12.*  Explain  the  use  of  ballast.     Q. 

I3.J   State    the    advantages   of  the    form    and   distribution   of 
weight  in  an  ordinary  hydrometer. 


68 


XLIV. 

FLOTATION. 

i.*  A  body  of  uniform  density  is  immersed  in  a  fluid  ;  state 
the  conditions  which  determine  whether  it  will  rise,  sink,  or 
remain  where  it  is,  when  free  to  move.  D.  155. 

2.J   Explain  the  action  of  the  "  Cartesian  diver."     D.  156. 

3.  Show  that  the  Cartesian  diver,    in   the  middle  of  a  fluid, 
is   never  in  stable  equilibrium,  as  far  as  height  is  concerned. 
D.  156. 

4.  Under  what  conditions  will  a  body,  denser  than  a  liquid, 
float  upon  its  surface  ? 

5.  Show   that   the   principle   of   the    last   question   can    be 
extended  to  the  case   of  bodies  supported  on   the  surface  of  a 
liquid  by  surface  tension.     D.  159. 

6.f  A  body  of  density,  d,  floats  on  a  liquid,  of  density,  D  ; 
find  the  proportion  of  its  volume  immersed. 

7.  A    body    of  density,    d,   and  volume,     V,    floats   with    a 
volume,    v,   immersed  in   a  liquid;  find  the   density,   D,  of  the 
liquid.     Prob.     Q. 

8.  A  body  floats  either  with  a   volume,   v,   immersed  in  a 
liquid  of  density,  D,  or  with  a  volume,    V,  immersed  in   a  liquid 
of  density,  d;  find  the  relative  densities  of  the  liquids  in  terms 
of  these  volumes.     Prob.     Q. 

9.J  Describe  the  construction  and  use  of  a  "specific  volume- 
nometer," (or  hydrometer  with  uniform  scale  showing  reciprocals 
of  density). 

10.  What,  in  general,  is  meant  by  a  hydrometer  of  variable 
immersion?  D.  168,  171. 

u.f  State  the  advantages  of  using  a  set  of  hydrometers 
instead  of  a  single  hydrometer,  covering  a  given  range  of 
density. 


69  FLOTATION.  [XLIV. 

1 2. |  Describe  a  ''densimeter"  (or  hydrometer  with  specific 
gravity  scale),  and  the  peculiar  spacing  of  the  scale  of  such  an 
instrument.  D.  172. 

13.  Name  certain  arbitrary  hydrometer  scales,  and  explain  the 
use  of  tables  in  connection  with  these  scales.     D.  173,  174. 

14.  State  the  advantages,  for  special  purposes,  of  instruments 
like  the  "  centesimal  alcoholimeter."     D.  175. 

15.  What  is  meant  by  a  hydrometer  of  constant  immersion? 
D.  168. 

1 6.  J   Describe    NICHOLSON'S    hydrometer,     and    the    general 
method  of  finding  by  it  the  weight  of  a  body  (in  air  or  in  water). 
D.  169. 

17.  Show  that   any  uniform  pressure  (whether   due   to  the 
atmosphere  or  not)  will   not  affect  the  equilibrium  of  a  hydro- 
meter.    Prob.     Q. 

1 8.  Show  that  any  cause  which  increases  the  pressure  on  the 
lower  part  of  a  hydrometer  more  than  on   the  upper  part  will 
have  a  buoyant  effect.     Prob.     Q. 

I9.J  Describe  and  explain  one  or  more  experiments  illustrating 
the  buoyant  effect  of  covering  the  upper  portion  of  a  hydrometer, 
floating  in  water,  with  a  liquid  of  lower  density. 

20. f  State  the  effect  of  atmospheric  density  on  a  hydrometer, 
and  the  advantage  of  reducing  the  dimensions  of  those  parts  of 
a  hydrometer  which  float  above  the  surface  of  a  liquid. 


XLV. 
PRINCIPLE  OF  ARCHIMEDES. 

1.  Prove  that  the  difference  between  the  forces  exerted  upon 
the  top   and  bottom   of   a  rigid  column  with   vertical    sides  of 
height,  /z,  including  an  area,  a,  by  a  fluid  of  density,  d,  must  be 
the  same  as  that  (hda)  upon  a  column  of  the  same  fluid  having 
identical   dimensions,    and   hence    equal    to    the    weight   of   the 
column  of  fluid.     D.  153. 

2.  Show  that  the  buoyant  action  of  a  fluid  upon   a  rigid 
column,  with  vertical  sides,  is  independent  of  the  depth.   D.  153. 

3.  Prove  (by  cutting  up  a  body  into  prismatic  columns  with 
vertical  sides)   that  any   body,    suspended  in    a  fluid,   must   be 
buoyed  up  with  a  force  equal  to  the  weight  of  an  equal  bulk  of 
the  fluid.     D.  153. 

4.  Prove  the  statement  in  the  last  question  by  substituting 
for  the  body,   a  body  of  the  fluid  having  the  same  shape,  and 
applying  the  general  conditions  of  equilibrium.     D.  153. 

5.  What  is  meant   by  the  loss  of  weight  of  a  body   when 
immersed  in  a  fluid  ?     D.  153. 

6.*  State  the  principle  of  Archimedes.     D.  153. 

7.  Show  that  the  principle  of  Archimedes  holds  for  any  fluid, 
whether  liquid  or  gaseous.  Prob. 

8. \  Explain  how  the  weight  of  a  body  in  water  can  be  found 
by  a  NICHOLSON'S  hydrometer,  (i)  if  its  density  is  greater,  and 
(2)  if  its  density  is  less  than  that  of  water.  (D.  169). 

9-t  When  is  the  weight  of  a  body  in  a  fluid  considered 
positive,  and  when  negative  ? 

io.f  Show  that  the  principle  of  Archimedes  applies  to  all 
bodies,  whether  their  density  be  greater  or  less  than  that  of  the 
fluid  in  which  they  are  immersed. 

1 1. 1  Describe  one  or  more  forms  of  hydrostatic  balance. 
D.  164. 


71  PRINCIPLE    OF  ARCHIMEDES.  [XLV. 

I2.J  Explain  the  use  of  a  sinker  in  weighing  bodies  in  a  liquid 
in  which  they  would  otherwise  float,  and  how  the  weight  of  the 
sinker  can  be  allowed  for.  D.  165. 

13. £  Describe  the  "overflow-beaker,"  and  its  use  for  finding 
the  weight  of  liquid  displaced  by  a  solid. 

I4.J  Show  that  the  weight  of  an  overflow  beaker  is  not 
increased  when  a  suspended  solid  is  immersed  in  it.  Prob. 

I5.J  Show  that  a  vessel  (without  overflow),  into  which  a 
suspended  solid  is  immersed,  gains  in  \veight  by  an  amount  equal 
to  the  weight  of  liquid  displaced  by  the  solid.  Prob. 

1 6.*  Prove  (by  considering  the  weight  of  a  solid,  as  borne 
partly  by  its  suspension,  and  partly  by  the  vessel  in  which  it  is 
immersed)  that  the  loss  of  weight  of  a  body  on  immersion  must 
necessarily  be  equal  to  the  gain  of  weight  of  the  vessel  in  which 
it  is  immersed.  Prob. 

i y.*t  Describe  one  or  more  experiments  illustrating  the 
principle  of  Archimedes.  D.  154. 

18.*  A  piece  of  chalk  weighs  (i)  5  grams  in  air  when  dry,  and 
(2)  2  grams  in  water ;  but  (3)  it  weighs  6  grams  in  air  when  the 
air  in  its  pores  has  been  completely  replaced  (through  long 
soaking)  by  water,  and  (4)  it  weighs  3  grams  under  the  same 
circumstances  in  water  ;  show  that  if  any  three  of  the  four  data 
above  are  given,  the  fourth  (assuming  the  chalk  to  be  inexpan- 
sible)  can  be  supplied.  Prob. 

iQ.f  Discuss  the  truth  of  the  statement  that  "water  weighs 
nothing  in  water." 

20.*  When  does  absorption  of  water  increase  the  weight  of  a 
body  in  water,  and  when  does  it  have  no  effect  ?  Q. 

2 1. *J Describe  one  or  more  experiments  with  a  rigid  vessel  and 
with  an  expansible  vessel,  illustrating  the  point  in  the  last 
question. 

22.  Why  was  ARISTOTLE'S  method  of  detecting  the  wreight 
of  air  unsuccessful  ?  and  how  should  it  have  been  modified  ? 
Prob.  (D.  195). 


72 


XLVI. 

ATMOSPHERIC  BUOYANCY. 

1.  About  what  is  the  weight  of  a  cubic  centimetre  of  air? 
of  a  litre?  of  a  cubic  metre?  etc.      (D.  195). 

2.  Calculate  the  lifting  power  of  a  balloon   of  weight,   w, 
volume,  v,  filled  with   gas  of  density,    d,    in   air  of  density,    D. 
D.  249. 

3.^   Describe  an  experiment  with  the  "  baroscope."     D.   247. 

4.J  Describe  a  "  barodeik  "  or  graduated  baroscope,  and 
explain  its  action. 

5-f  What  is  meant  by  the  "  effective  weight"  of  a  body  in  a 
fluid? 

6.  Find  the  effective  weight  (in  gravitation  units)  of  a  body 
of  mass,  M,  and  density,  D,  in  air  of  density,  d.     Prob.     Q. 

7.  Find  the   effective  weight   (in   gravitation  units)    of   M 
grams  of  brass  of  density  8.4   in  air  of  density  0.0012  (grams  to 
the  cubic  centimetre).     Prob.     Q. 

8.  Under    what    circumstances    are    standards    of    weight 
supposed  to  be  adjusted  ?     0.251. 

9-f  Show  that  all  ordinary  weighings  consist  in  balancing 
effective  weights  in  air. 

io.f  What  is  meant  by  the  "apparent  weight"  of  a  body  (in 
air)  ?  D.  251. 

n.f  Show  that  the  effective  weight  of  a  body  is  always  less 
than  its  apparent  weight  by  a  certain  proportional  amount  (about 
i  part  in  7000)  depending  only  upon  the  density  of  the  air  (say 
0.0012)  and  of  the  brass  weights  (say  8.4)  by  which  it  is 
counterpoised.  D.  i. 

12.  Given  the  apparent  weight,  W,  and  density,  D,  of  a  body 
counterpoised  by  brass  standards  of  density,  D' ,  in  air  of  density, 
d,  find  the  true  weight  of  the  body  in  vacuo.  D.  251. 


73  ATMOSPHERIC    BUOYANCY.  [XLVI. 

13.     When  is  the  correction  for  buoyancy  positive   and  when 
negative?     Prob.     Q.  n. 

i4-t  Illustrate  the  corrections  for  air-buoyancy  by  numerical 
examples. 

15.  Show  that  the  apparent  weight  of  a  body  in   air  varies 
according  to  conditions  of  atmospheric  density,  and  that,  therefore, 
all  exact    weighings  should  be  reduced  to  vacuo.     Prob.      (D. 
160,  251).     Q.  ii. 

1 6.  Prove  that  the  effective  weight  of  a  body  in  water  is  less 
than   in  air  by  an   amount  equal  to  the  difference  between  the 
weights  of  water  and  of  air  displaced,    and  hence  equal  to  the 
effective  weight  of  the  water  displaced.     Prob.     Q. 

17.  Prove  (by  considering  the  relation  between   effective  and 
apparent  weights)  that  the  statement  in  the  last  question  applies 
to  apparent  as  well  as  to  effective  weights.     Prob.     Q.  n,  14. 

1 8.  Show  that  the  principle  of  Archimedes  applies  to  effective 
or  to  apparent  weights,  as  well  as  to  true  weights.     Prob.     Q. 


74 


XLVIL 
APPARENT  SPECIFIC  GRAVITY. 

i.f  What  name  is  given  to  the  ratio  between  the  apparent 
weight  of  a  given  substance  and  the  apparent  weight  of  an 
equal  volume  of  water,  or  other  standard  of  reference  ? 

2.*  Define  "  specific  gravity,"  and  distinguish  it  from  density. 
D.  1 60. 

3-f  Define  "apparent  specific  gravity,"  and  distinguish  it 
from  density  as  well  as  from  true  specific  gravity. 

4.f  What  substance,  or  substances,  are  chosen  as  standards 
of  reference  for  specific  gravity  (i)  in  the  case  of  liquids  or  solids 
and  (2)  in  the  case  of  gases  ?  (D.  160). 

5.f  Criticise  the  common  use  of  the  term  "vapor  density" 
(as  synonymous  with  specific  gravity). 

6.f  At  what  temperature  did  the  founders  of  the  metric  system 
intend  that  the  density  of  water  should  be  exactly  i,  and  at 
about  what  two  temperatures  is  this  condition  new  thought  to  be 
fulfilled?  (D.  1 60). 

y.f  Give  some  idea  of  the  density  of  water  at  ordinary 
temperatures,  and  of  the  delicacy  of  the  apparatus  necessary  to 
detect  variations  in  its  density. 

8.  Given  that  a  cubic  decimetre  of  wood  weights  499  grams 
in  air,  and  500  grams  in  vacuo  ;  also   that  a  cubic  decimetre  of 
water  at  a  certain  temperature  weighs  997  grams  in  air,  and   998 
grams  in  vacuo  ;  find   the  density  of  the  wood,  and  its  true  and 
apparent  specific  gravities  referred  to  water  at  the  given  tempera- 
ture.    Prob.     Q. 

9.  *£  Explain  the  determination  of  (apparent)  specific  gravity 
by  means  of   a   (Joiyi<Y)   spring   balance,    or   by    a    hydrostatic 
balance.     D.  164. 


75  APPARENT  SPECIFIC   GRAVITY.  [XLVII. 

10.  Given  the  (apparent)  weights,  W  and  w,  of  a  body  in  air 
and   in  water,  how  do  you  find  the  (apparent)  specific  gravity  of 
the  body?     D.  164. 

11.  If   W  is  the  (apparent)  weight  of  a  body  in  air,  how  do 
you  find  its  (apparent)  specific  gravity  if  w  is  (i)  the  (apparent) 
loss   of  weight  of  the   body   when  weighed  in   water  ?    (2)  the 
(apparent)  gain  in  weight  of  the  vessel  in  which  it  is  immersed? 
or  (3)  the  (apparent)  weight  of  the  water  displaced  ?     (D.  164). 

i2*fA  piece  of  dry  chalk  weighs  5  grams  in  air,  and  2  grams 
in  water,  but  it  takes  i  gram  of  water  to  replace  all  the  air  in  its 
pores  ;  find  the  (apparent)  specific  gravity  (i)  of  the  dry  chalk, 
(2)  of  the  wet  chalk,  and  (3)  of  the  material  within  which  the 
pores  of  the  chalk  are  included. 

13.  How  do  you  find  the  (apparent)  specific  gravity  of  a  liquid, 
if  W^and  w  are  respectively  (i)  the  (apparent)  losses  of  weight 
of  a  given  body  when  immersed  in  water  and  in  the  liquid,  (2) 
the  corresponding  (apparent)  gains  in  weight  on  the  part  of  the 
vessel,  or  (3)  the  (apparent)  weights  of  water  and  of  liquid 
displaced  by  a  given  body  ?  Prob.  (D.  166). 

14.*! Explain  the  determination  of  (apparent)  specific  gravity 
by  means  of  a  specific  gravity  bottle.  (D.  163). 

15.*  A  vessel  is  filled  by  an  (apparent)  weight,  W,  of  water  or 
by  an  (apparent)  weight,  w,  of  a  given  liquid  ;  find  the 
(apparent)  specific  gravity  of  the  liquid.  (D.  163). 


76 


XLVIII. 
LAW  OF  AVOGADRO. 

i.f  What  is  meant  by  the  molecular  constitution  of  matter? 

2.f  State  the  law  of  chemical  combining  proportions  of  gases 
in  terrns  of  their  volumes. 

3.f  What  conclusions  have  Chemists  adopted  in  regard  to  the 
number  of  molecules  in  a  given  volume  of  different  gases  at  the 
same  pressure  and  temperature  ? 

4.f  State  the  law  of  Avogadro. 

5.f  What  is  meant  by  the  molecular  weight  of  a  gas  ? 

6.f  Find  the  relation  between  the  molecular  weight  of  a  gas 
and  its  specific  gravity  referred  to  hydrogen,  assuming  that  the 
molecular  weight  of  hydrogen  is  2. 

7.  Given  the  density  of  a  gas,  how  do  you  find  its  molecular 
weight  ?  Illustrate  by  the  following  examples  : 

Density  of  hydrogen    (o°  and  76  cm.)      0.00008952 
"        "    oxygen  0.0014291 

"       "    nitrogen  0.0012544 

Prob.     Q. 

8.J  Explain  one  or  more  methods  (e.g.,  DUMAS')  for  deter- 
mining the  molecular  weight  of  a  gas  (or  vapor).. 


77 


XLIX. 

MOLECULAR  THEORY  OF  THE  PRESSURE  OF  GASES. 

i.f  Through  what  agency  is  the  pressure  of  a  gas  believed  to 
be  maintained  ? 

2.f  Show  that,  if  pressure  is  maintained  by  molecular  impact, 
it  should  vary  as  the  number  of  molecules  in  a  given  space, 
provided  that  the  molecules  are  not  so  near  together  as  to 
interfere  with  one  another. 

3.  State  DALTON'S  law  for  gaseous  mixtures,  and  its  applica- 
bility in  certain  cases  to  mixtures  of  vapors  and  gases.  D.  227. 

4.J  Describe  one  or  more  experiments  illustrating  DALTON'S 
law. 

5.f  What  necessary  relation  exists  between  the  number  of 
molecules  in  a  given  space  and  the  density  of  a  gas  ? 

6.f  Show  the  pressure  of  a  gas,  if  sufficiently  rare,  must  vary 
as  its  density. 

y.f  State  results  of  experimental  investigation  as  to  the  truth 
of  the  statement  in  the  last  question. 

8.  What  conclusion  did  DULONG  and  ARAGO  arrive  at  with 
respect  to  the  compressibility  of  air?  D.  219. 

9.f  Within  what  limits  have  the  results  of  DULONG  and 
ARAGO  been  confirmed  by  later  experiments?  (D.  219). 

10.  f  Name  certain  gases  in  which  the  density  is  found  to  be  at 
least  approximately  proportional  to  the  pressure  for  variations  of 
one  or  two  atmospheres. 


L. 
LAW  OF  BOYLE  AND  MARIOTTE. 

1.  State  the  necessary  relation  between   the  density  and  the 
volume  of  a  given  mass  of  gas.     Prob. 

2.  Prove  that,   according    to  the    molecular    theory    of   the 
pressure  of  a  gas,  the  pressure  must  vary  inversely  as  the  volume. 
Prob.     Q. 

3.*  State  the  law  of  BOYLE  and  MARIOTTE.     D.  217. 

4.  Show  that  the  product,  VP,  of  the  volume,  V,  and  the 
pressure,  P,  of  a  given  mass  of  gas,  obeying  the  law  of  BOYLE 
and  MARIOTTE,  is  constant.  Prob.  Q. 

5.|  Describe  an  experiment  with  BOYLE'S  or  MARIOTTE'S 
tube,  illustrating  the  law  of  BOYLE  and  MARIOTTE.  D.  218. 

6.J  Describe  an  experiment  with  a  mercury  well,  showing 
that  the  law  of  BOYLE  and  MARIOTTE  holds  for  pressures  less 
than  that  of  the  atmosphere.  D.  218. 

7,J  Describe  one  or  more  experiments  with  a  piece  of 
apparatus  capable  of  illustrating  the  law  of  BOYLE  and  MARIOTTE 
for  pressures  both  greater  and  less  than  that  of  the  atmosphere. 

8.  Describe  and  explain  the  use  of  one  or  more  forms  of 
•compressed  air  manometer.  D.  225. 


79 


LI. 
UNEQUAL  COMPRESSIBILITY  OF  GASES. 

i.f  What  limit  would  the  volume  of  an  ideal  gas  approach  if 
the  pressure  should  become  indefinitely  great  ?  and  how  would 
this  limit  be  modified  in  practice  if  the  molecules  of  the  gas  were 
incompressible,  and  occupied  a  certain  amount  of  space  ? 

2.f  Show  that  the  size  of  molecules  would  diminish  the 
compressibity  of  a  gas. 

3.f  What  would  be  the  effect  of  attractive  forces  between  the 
molecules  of  a  gas  on  the  volume  occupied  at  a  given  pressure  ? 

4.f  Show  that  forces  of  the  nature  of  adhesion  or  cohesion 
would  tend  to  increase  the  compressibility  of  a  gas. 

5.  Explain  the  departure  of  certain  gases  from  the  law  of 
BOYLE  and  MARIOTTE.     Prob.     Q. 

6.  Describe    one  or  more  experiments  (due    to    DKSPRETZ, 
POUILLET,  or  REGNAULT)  showing  the  unequal  compressibility 
of  different  gases.     D.  219,  220,  221. 

y.f  Give  some  idea  of  the  percentage  error  in  calculating  the 
density  or  pressure  of  the  more  permanent  gases,  by  the  law  of 
BOYLE  and  MARIOTTE,  within  limits  of  one  or  two  atmospheres. 


8o 


LII. 
VAPORS  AND  GASES  DISTINGUISHED. 

i.J   Describe   the   effects   of    increasing   pressure    upon    the 
volume  of  a  vapor. 

2.f  What  is  meant  by  the  "maximum  pressure"  (or tension) 
of  a  vapor  ? 

3-t  What  is  meant  by  the  condensation  of  a  vapor  ? 
4-t  Distinguish  vapors  from  gases  in  respect  to  phenomena  of 
condensation. 

5.f  Distinguish  between  the  condensation  of  a  vapor  and  the 
absorption  of  a  vapor  by  a  liquid. 

6.     What  is  meant   by  the  solubility  of  a  gas  or  vapor  in  a 
liquid  at  a  given  pressure  and  temperature  ?     D.  228. 

y.f  State  the  law  connecting  the  solubility  of  a  gas  or  vapor 
in  a  liquid  with  the  pressure. 

8.     Describe  one  or  more  experiments  illustrating  the  absorp- 
tion of  gases  or  vapors  by  charcoal.     D.  228. 

9.f  What  is  meant  by  the  occlusion  of  a  gas? 

10. J  Describe  one  or  more  experiments  illustrating  the 
occlusion  of  hydrogen  by  platinum  oriridium.  (D.  228). 

n.J  Describe  one  or  more  experiments  showing  that  a  film  of 
air  adheres  to  glass  or  other  solids  under  certain  conditions. 
(D.  228). 

i2.f  What  connection  probably  exists  between  the  causes 
giving  rise  to  phenomena  of  condensation,  absorption,  occlusion, 
solubility,  etc.,  in  certain  gases  or  vapors,  and  their  departure 
from  the  law  of  BOYLE  and  MARIOTTE  ? 


END  OF  PART  I. 

'THTI7BRSIT71 


far--,  =Ts":===s*===^ 


YC   11421 


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